Methods of using combined forward and backward sampling of nuclear magnetic resonance time domain for measurement of secondary phase shifts, detection of absorption mode signals devoid of dispersive components, and/or optimization of nuclear magnetic resonance experiments

ABSTRACT

The present invention relates to a method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by the use of forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts. The present invention also relates to methods of conducting an N-dimensional NMR experiment in a phase-sensitive manner by the use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in the frequency domain from secondary phase shifts.

This application claims the benefit of U.S. Provisional PatentApplication Ser. No. 61/028,070, filed Feb. 12, 2008 and U.S.Provisional Patent Application Ser. No. 61/092,901, filed Aug. 29, 2008,which are hereby incorporated by reference in their entirety.

The subject matter of this application was made with support from theUnited States Government under The National Institutes of Health GrantNo. U54 GM074958-01 and the National Science Foundation Grant Nos. MCB0416899 and MCB 0817857. The government has certain rights in thisinvention.

FIELD OF THE INVENTION

The present invention relates to a method of conducting an N-dimensionalnuclear magnetic resonance (NMR) experiment in a phase-sensitive mannerby the use of forward and backward sampling of time domain shifted by aprimary phase shift under conditions effective to measure time domainamplitudes and secondary phase shifts. The present invention alsorelates to methods of conducting an N-dimensional NMR experiment in aphase-sensitive manner by the use of dual forward and backward samplingof time domain shifted by a primary phase shift under conditionseffective to at least partially cancel dispersive and quadrature imagesignal components arising in the frequency domain from secondary phaseshifts.

BACKGROUND OF THE INVENTION

Multi-dimensional Fourier Transform (FT) NMR spectroscopy is broadlyused in chemistry (Ernst et al., “Principles of Nuclear MagneticResonance in One and Two Dimensions,” Oxford: Oxford University Press(1987); Jacobsen, N. E., “NMR Spectroscopy Explained,” Wiley, New York(2007)) and spectral resolution is pivotal for its performance. The useof forward and backward sampling for pure absorption mode signaldetection to obtain improved spectral resolution is described in Bachmanet al., J. Mag. Res., 28:29-39 (1977), however, this methodology doesnot allow phase-sensitive detection. In particular, Bachmann et al.teach the combined forward-backward sampling of a chemical shiftevolution along one axis, the ‘x-axis’, only. This results in two cosinemodulations which, after addition, yield a real time domain signaldevoid of terms that lead to dispersive components in the frequencydomain spectrum obtained after a cosine transformation. Hence, Bachmannet al. do not teach phase-sensitive detection of chemical shifts.

Phase-sensitive, pure absorption mode signal detection (Ernst et al.,“Principles of Nuclear Magnetic Resonance in One and Two Dimensions,”Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMRSpectroscopy,” 2nd Ed., San Diego Academic Press (2007); Schmidt-Rohr etal., “Multidimensional Solid-State NMR and Polymers,” New York: AcademicPress (1994)) is required for achieving high spectral resolution sincean absorptive signal at frequency Ω₀ rapidly decays proportional to1/(Ω₀−Ω)² while a dispersive signal slowly decays proportional to1/(Ω₀−Ω). Hence, a variety of approaches were developed to accomplishpure absorption mode signal detection (Ernst et al., “Principles ofNuclear Magnetic Resonance in One and Two Dimensions,” Oxford: OxfordUniversity Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,”2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al.,“Multidimensional Solid-State NMR and Polymers,” New York: AcademicPress (1994)). Moreover, by use of techniques such as spin-lock purgepulses (Messerle et al., J. Magn. Reson. 85:608-613 (1989)), phasecycling, (Ernst et al., “Principles of Nuclear Magnetic Resonance in Oneand Two Dimensions,” Oxford: Oxford University Press (1987)) pulsedmagnetic field gradients, (Keeler et al., Methods Enzymol. 239:145-207(1994)) or z-filters (Sorensen et al., J. Magn. Reson. 56:527-534(1984)), radio-frequency (r.f.) pulse sequences for phase-sensitivedetection are designed to avoid ‘mixed’ phases, so that only phaseerrors remain which can then be removed by a zero- or first-order phasecorrection. A limitation of the hitherto developed approaches (Ernst etal., “Principles of Nuclear Magnetic Resonance in One and TwoDimensions,” Oxford: Oxford University Press (1987); Cavanagh et al.,“Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007);Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,”New York: Academic Press (1994)) arises whenever signals exhibit phaseerrors which cannot be removed by a zero- or first-order correction, orwhen aliasing limits (Cavanagh et al., “Protein NMR Spectroscopy,” 2ndEd., San Diego: Academic Press (2007)) first-order phase corrections to0° or 180°. Due to experimental imperfections, such phase errorsinevitably accumulate to some degree during the execution of r.f. pulsesequences (Ernst et al., “Principles of Nuclear Magnetic Resonance inOne and Two Dimensions,” Oxford: Oxford University Press (1987);Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego:Academic Press (2007); Schmidt-Rohr et al., “MultidimensionalSolid-State NMR and Polymers,” New York: Academic Press (1994)) whichresults in superposition of the desired absorptive signals withdispersive signals of varying relative intensity not linearly correlatedwith Ω₀. This not only exacerbates peak identification, but also reducesthe signal-to-noise (S/N) and shifts the peak maxima. In turn, thisreduces the precision of chemical shift measurements and impedesspectral assignment based on matching of shifts.

Furthermore, phase-sensitive, pure absorption mode detection of signalsencoding linear combinations of chemical shifts relies on joint samplingof chemical shifts as in Reduced-dimensionality (RD) NMR (Szyperski etal., J. Am. Chem. Soc. 115:9307-9308 (1993); Brutscher et al., J. Magn.Reson., B109:238-242 (1995); Szyperski et al., Proc. Natl. Acad. Sci.USA 99:8009-8014 (2002)) and its generalization, G-matrix Fouriertransform (GFT) projection NMR (Kim et al., J. Am. Chem. Soc.125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004);Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al.,J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol.394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A.102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692(2007)). The latter is broadly employed, in particular also (Szyperskiet al., Magn. Reson. Chem. 44:51-60 (2006)) forprojection-reconstruction (PR) (Kupce et al., J. Am. Chem. Soc.126:6429-6440 (2004); Coggins et al., J. Am. Chem. Soc. 126:1000-1001(2004)), high-resolution iterative frequency identification (HIFI)(Eghbalnia et al., J. Am. Chem. Soc. 127:12528-12536 (2005)), andautomated projection (APSY) NMR (Hiller et al., Proc. Natl. Acad. Sci.U.S.A. 102:10876-10881 (2005)). Importantly, the joint sampling ofchemical shifts entangles phase errors from several shift evolutionperiods. Hence, zero- and first-order phase corrections cannot beapplied in the GFT dimension (Atreya et al., J. Am. Chem. Soc.129:680-692 (2007)), which further accentuates the need for approacheswhich are capable of eliminating (residual) dispersive components.

In Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007), measurement ofnuclear spin-spin coupling is taught. In particular, Atreya et al. teachtransforming a secondary phase shift in the cosine J-modulation arisingfrom ‘J-mismatch’, that is, variation of J by spins system requiringdifferent Δt=½J delays for each spin system, into an imbalance ofamplitudes of sine and cosine modulation. This yields quadrature imagepeaks which are not removed. Atreya et al. teach combined forward andbackward sampling of cosine modulations, since sine modulations are notaffected by secondary phase shifts arising from J-mismatch.

The present invention is directed to overcoming these and otherdeficiencies in the art.

SUMMARY OF THE INVENTION

The present invention relates to a method of conducting an N-dimensionalNMR experiment in a phase-sensitive manner by use of forward (from time0 to +t) and backward (from time 0 to −t) sampling of time domainshifted by a primary phase shift under conditions effective to measuretime domain amplitudes and secondary phase shifts. The method comprisesproviding a sample; applying radiofrequency pulses for an N-dimensionalNMR experiment to said sample; selecting m dimensions of said NMRexperiment, wherein m≦N, sampling a time domain modulation in aphase-sensitive manner in each selected dimension jε[1,2, . . . , m]arising from time evolution of chemical shift α_(j) in both a forwardand backward manner to obtain two interferograms for each time domaindimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},{wherein}}\end{matrix}$

I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j,δ) _(j) ⁻ are amplitudes, Ψ_(j) andΨ_(j)+δ_(j) are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[ and thecases {ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 beingomitted, and Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻ are secondaryphase shifts; multiplying each said vectors C_(j,ψ) _(j) (t_(j)) with aD-matrix defined as

$D_{j} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}$

and a vector Q=[1 i], wherein i=√{square root over (−1)}, according toQ·D_(j)·C_(j,ψ) _(j) (t_(j)) under conditions effective to create acomplex time domain of said selected m dimensions according to

${\underset{j}{\otimes}{Q \cdot D_{j} \cdot {C_{j,\psi_{j}}\left( t_{j} \right)}}};$

and transforming said complex time domain into frequency domain by useof an operator O under conditions effective to measure the values ofI_(j,ψ) _(j) ⁺, I_(j,ψ) _(j) ⁻, Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j) ⁻ insaid frequency domain.

Another aspect of the present invention relates to a method ofconducting an N-dimensional NMR experiment in a phase-sensitive mannerby use of dual forward (from time 0 to +t) and backward (from time 0 to−t) sampling of time domain shifted by a primary phase shift underconditions effective to measure secondary phase shifts or at leastpartially cancel dispersive and quadrature image signal componentsarising in a frequency domain from secondary phase shifts. The methodcomprises providing a sample; applying radiofrequency pulses for anN-dimensional NMR experiment to said sample; selecting m dimensions ofsaid NMR experiment, wherein m≦N, sampling a time domain modulation in aphase-sensitive manner in each said selected dimension jε[1, 2, . . . ,m] arising from time evolution of chemical shift α_(j) in both a forwardand backward manner to obtain two interferograms for each time domaindimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},{wherein}}\end{matrix}$

I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j) ⁻ are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j)are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[ and the cases{ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted,and Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j) ⁻ are secondary phase shifts;multiplying each said vectors C_(j,ψ) _(j) (t_(j)) with a D-matrixdefined as

$D_{j} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}$

and a vector Q=[1 i], wherein i=√{square root over (−1)}, according toQ·D_(j)·C_(j,ψ) _(j) (t_(j)) under conditions effective to create acomplex time domain of said selected m dimensions according to

${\underset{j}{\otimes}{Q \cdot D_{j} \cdot {C_{j,\psi_{j}}\left( t_{j} \right)}}};$

repeating said selecting, said sampling and said multiplying(2^(m))-times, thereby sampling the m dimensions with all 2^(m) possiblepermutations resulting from selecting for each dimension j either Ψ_(j)or Ψ_(j)+π/2, with δ_(j) being incremented by either 0 or π, therebyyielding 2^(m) complex time domains; linearly combining said 2^(m)complex time domains; and transforming said linearly combined complextime domain into frequency domain by use of an operator O, underconditions effective to at least partially cancel dispersive andquadrature image peak components arising from Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ)_(j) ⁻ in said frequency domain.

A further aspect of the present invention relates to a method ofconducting an N-dimensional NMR experiment in a phase-sensitive mannerby use of dual forward (from time 0 to +t) and backward (from time 0 to−t) sampling of time domain shifted by a primary phase shift underconditions effective to measure secondary phase shifts or at leastpartially cancel dispersive and quadrature image signal componentsarising in a frequency domain from secondary phase shifts. The methodcomprises providing a sample; applying radiofrequency pulses for anN-dimensional NMR experiment to said sample; selecting m dimensions ofsaid NMR experiment, wherein m≦N, sampling twice a time domainmodulation in a phase-sensitive manner in each said selected dimensionjε[1,2, . . . , m] arising from time evolution of chemical shift α_(j),once in a forward manner to obtain two interferograms for each timedomain dimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}^{+}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{c_{\psi_{j},\delta_{j}}^{+}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{\cos \left( {\psi_{j} + \delta_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}}\end{bmatrix}},{and}}\end{matrix}$

once in a backward manner to obtain two interferograms for each timedomain dimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}^{-}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{-}{c_{\psi_{j}}^{-}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{-}{\cos \left( {\psi_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{-}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},}\end{matrix}$

wherein I_(j,ψ) _(j) ⁺, I_(j,ψ) _(j) _(,δ) _(j) ⁺, I_(j,ψ) _(j) ⁻, andI_(j,ψ) _(j) _(,δ) _(j) ⁻ are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j) areprimary phase shifts with Ψ_(j), δ_(j)ε[0,2π[, and Φ_(j,ψ) _(j) ⁺,Φ_(j,ψ) _(j) _(,δ) _(j) ⁺, Φ_(j,ψ) _(j) ⁻, and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻are secondary phase shifts; multiplying each said vector C_(j,ψ) _(j)⁺(t_(j)) with a D-matrix defined as

$D_{j}^{-} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{\cos \left( {\psi_{j} + \delta_{j}} \right)} & {- {\cos \left( \psi_{j} \right)}}\end{bmatrix}$

and each said vector C_(j,ψ) _(j) ⁻(t_(j)) with a D-matrix defined as

${D_{j}^{-} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}};$

multiplying the said products D_(j) ⁺·C_(j,ψ) _(j) ⁺(t_(j)) and D_(j)⁻·C_(j,ψ) _(j) ⁻(t_(j)) with a vector Q=[1 i], wherein i=√{square rootover (−1)}, according to Q·D_(j) ⁺·C_(j,ψ) _(j) ⁺(t_(j)) and Q·D_(j)⁻·C_(j,ψ) _(j) ⁻(t_(j)) under conditions effective to create a complextime domain of said selected m dimensions according to

${\underset{j}{\otimes}{{Q \cdot D_{j}^{-} \cdot {C_{j,\psi_{j}}^{+}\left( t_{j} \right)}}\mspace{14mu} {{{and}\mspace{11mu} \underset{j}{\otimes}Q} \cdot D_{j}^{-} \cdot {C_{j,\psi_{j}}^{-}\left( t_{j} \right)}}}};$

repeating said selecting, said phase-sensitive sampling twice and saidmultiplying (2^(m)−2)-times, thereby sampling said m dimensions with all2^(m) possible permutations resulting from selecting for each dimensionj either phase-sensitive forward or backward sampling according toC_(j,ψ) _(j) ⁺(t_(j)) or C_(j,ψ) _(j) ⁻(t_(j)); linearly combining said2^(m) complex time domains; and transforming said linearly combinedcomplex time domain into frequency domain by use of an operator O, underconditions effective to at least partially cancel dispersive andquadrature image peak components arising from Φ_(j,ψ) _(j) ⁺, Φ_(j,ψ)_(j) _(,δ) _(j) ⁺, Φ_(j,ψ) _(j) ⁻, and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻, insaid frequency domain.

Pure absorption mode NMR spectra in accordance with the presentinvention are most amenable to automated (Moseley et al., J. Magn.Reson. 170:263-277 (2004); Lopez-Mendez et al., J. Am. Chem. Soc.128:13112-13122 (2006), which are hereby incorporated by reference intheir entirety) peak ‘picking’ and the resulting increased precision ofshift measurements also increases the efficiency of automated resonanceassignment of NMR spectra (Moseley et al., Methods Enzymol. 339:91-108(2001), which is hereby incorporated by reference in its entirety). Thisis because chemical shift matching tolerances can be reduced (For theprogram AutoAssign (Moseley et al., Methods Enzymol, 339:91-108 (2001),which is hereby incorporated by reference in its entirety), a matchingtolerance of 0.3 ppm is routinely used for ¹³C^(α/β) chemical shifts.Since shifts of peak maxima are up to about ±0.07 ppm, elimination ofdispersive components enables one to reduce the tolerancesignificantly). Moreover, the enhanced spectral resolution promises tobe of particular value for systems exhibiting very high chemical shiftdegeneracy such as (partially) unfolded or membrane proteins.

Pure absorption mode NMR data acquisition as set forth in the presentinvention enables one to also remove dispersive components arising fromsecondary phase shifts, such as phase errors, which cannot be removed bya zero- or first-order phase correction. Hence, such data acquisitionresolves a long-standing challenge of both conventional (Ernst et al.,“Principles of Nuclear Magnetic Resonance in One and Two Dimensions,”Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMRSpectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohret al., “Multidimensional Solid-State NMR and Polymers,” New York:Academic Press (1994), which are hereby incorporated by reference intheir entirety) and GFT-based projection NMR (Kim et al., J. Am. Chem.Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004);Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al.,J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol.394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A.102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692(2007); Szyperski et al., Magn. Reson. Chem. 44:51-60 (2006); Kupce etal., J. Am. Chem. Soc. 126:6429-6440 (2004); Coggins et al., J. Am.Chem. Soc. 126:1000-1001 (2004); Eghbalnia et al., J. Am. Chem. Soc.127:12528-12536 (2005); Hiller et al., Proc. Natl. Acad. Sci. U.S.A.102:10876-10881 (2005), which are hereby incorporated by reference intheir entirety). Furthermore, the present invention promises to broadlyimpact NMR data acquisition protocols for science and engineering.

The present invention relates to phase-sensitive detection of chemicalshifts, which is reflected by the fact that the cases {ψ_(j)=nπ/2 andδ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 are omitted in the claims ortime-proportional phase incrementation (TPPI) is used. This is incontrast to the prior art, for example, Bachmann et al. J. Mag. Res.,28:29-39 (1977) (with ψ_(j)=δ_(j)=0), which does not relate tophase-sensitive detection of chemical shifts.

Moreover, in contrast to the prior art, the present invention teacheshow secondary phase shifts can be measured in frequency domain spectrain which chemical shifts are measured, and how frequency domain signalsor signal components arising from the secondary shifts can be partly orentirely canceled. Nuclear spin-spin couplings (as measured in Atreya etal., J. Am. Chem. Soc. 129:680-692 (2007), which is hereby incorporatedby reference in its entirety) and chemical shifts (as measured in thepresent invention) result from fundamentally different interactions,that is, from interaction among nuclear spins and interaction of nuclearspins with the magnetic field, respectively, and thus result in afundamentally different types of time evolution of spin systems.Considering a two-spin system containing spins I and S, time evolutioncan be conveniently described by use of corresponding Cartesian spinoperators I and S (Cavanagh et al., “Protein NMR Spectroscopy,” SanDiego: Academic Press, 2^(nd) Ed., (2007), which is hereby incorporatedby reference in its entirety). The time evolution of transversemagnetization I_(x) present at time t=0 is given by:

$\begin{matrix}{{{{I_{x}\overset{t}{}I_{x}}{\cos \left( {\omega \; t} \right)}} + {I_{y}{\sin \left( {\omega \; t} \right)}}},} & ({A1})\end{matrix}$

revealing that magnetization aligned along the x-axis at time t=0evolves into the magnetization pointing along the orthogonal y-axis. ForI_(y) one obtains:

$\begin{matrix}{{{I_{y}\overset{t}{}I_{y}}{\cos \left( {\omega \; t} \right)}} - {I_{x}{{\sin \left( {\omega \; t} \right)}.}}} & ({A2})\end{matrix}$

In contrast, time evolution driven by nuclear spin-spin couplings, e.g.J-couplings, is given by

$\begin{matrix}{{{{I_{x}\overset{t}{}I_{x}}{\cos \left( {\pi \; {Jt}} \right)}} + {2\; I_{y}S_{z}{\sin \left( {\pi \; {Jt}} \right)}}},} & ({A3})\end{matrix}$

where in-phase magnetization is converted into anti-phase magnetization,and

$\begin{matrix}{{{I_{y}{S_{z}\overset{t}{}I_{y}}S_{z}{\cos \left( {\pi \; {Jt}} \right)}} - {I_{x}{\sin \left( {\pi \; {Jt}} \right)}}},} & ({A4})\end{matrix}$

where anti-phase magnetization is converted into in-phase magnetization.

Comparison of equations (A1) and (A2) with (A3) and (A4) reveals that astraightforward phase-sensitive detection of J-coupling evolution is notpossible. Instead, sine modulation must obtained when starting with atype of spin state, that is, anti-phase magnetization which needs to becreated during a delay Δt=½J.

As pointed out by Freeman and Kupce (Freeman et al., Concepts in Mag.Res. 23A:63-75 (2004), which is hereby incorporated by reference in itsentirety), experiments measuring nuclear spin-spin couplings (as inAtreya et al., J. Am. Chem. Soc. 129:680-692 (2007), which is herebyincorporated by reference in its entirety) thus belong to the class ofNMR experiments in which a ‘real variable’ is sampled, while chemicalshift evolution, as in the present invention, is sampled in a complexmanner when performed phase-sensitively.

The present invention teaches measurement of secondary phase shiftsand/or partial or entire elimination of dispersive frequency domaincomponents, quadrature image frequency domain peaks, as well ascross-talk peaks in GFT NMR spectra. In particular, the presentinvention teaches implementation of combined phase-shifted forward andbackward sampling of chemical shifts. Due to the fundamental differenceof time evolution of transverse magnetization under chemical shift andJ-coupling, the implementation of sampling of corresponding timeevolution in NMR experiments are different. This becomes apparent wheninspecting the radio-frequency (r.f) pulse schemes effective toimplement J-GFT NMR experiments or NMR experiments designed to measurechemical shifts. In order to phase-sensitively detect chemical shift,phases of r.f. pulses creating transverse magnetization are incremented.In contrast, detection of cosine and sine J-modulations requires that180° r.f. pulse, which can refocus the evolution arising fromJ-couplings, are shifted in time.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-B illustrate clean absorption mode NMR data acquisition.

FIG. 1A shows dual ‘States’ and FIG. 1B shows (c⁻¹,c⁻¹,c₊₃,c⁻³)-dualphase shifted mirrored sampling (DPMS). The residual phase error Φ isassumed to be 15°. For comparison of the intensity and phase of theinterferograms, the dashed grey lines represent time domain data of unitamplitude and Φ=0. In FIG. 1A, the intensities of frequency domain peakswere calculated using the following equation:

States = I 0 States  ( Φ ) I 0 States  ( Φ = 0 ) = sin 2  Φ 2   R 2 ( 1 - cos   Φ ) / 1 R 2 = sin 2  Φ 2  ( 1 - cos   Φ )

FIG. 2 illustrates percentage reduction of signals' maximum for ‘States’(black upper line) and (c₊₁,c⁻¹)-PMS (phase-shifted mirrored time domainsampling) (dashed lower line) data acquisition versus Φ. For Φ=15°, thereduction arising from PMS is approximately 1.7% larger than for‘States’.

FIGS. 3A-B illustrate a ¹³C r.f. pulse module enabling forward (FIG. 3A)and backward (FIG. 3B) sampling of the ¹³C chemical shift evolutionperiod in 2D [¹³C,¹H]-HSQC (Cavanagh et al., “Protein NMR Spectroscopy,”2nd Ed., San Diego: Academic Press (2007), which is hereby incorporatedby reference in its entirety). Filled and open bars represent,respectively, 90° and 180° pulses. Pulse phases are indicated above the90° pulses, and the phase of the 180° pulse is 0. The setting of φdefines the type of sampling. For c₊₀ or c⁻⁰-interferograms, φ=−π/2; Forc₊₂ or c⁻²-interferograms, φ=0; For c₊₀ or c⁻⁰-interferograms, φ=−π/4;For c₊₃ or c⁻³-interferograms, φ=π/4

FIGS. 4A-I illustrate cross sections taken along ω₁(¹³C) of 2D[¹³C,¹H]-HSQC spectra recorded with (c₊₀,c₊₂)-sampling (FIG. 4A),(c⁻⁰,c⁻²)-sampling (FIG. 4B), (c₊₀,c₊₂,c⁻⁰,c⁻²)-sampling (FIG. 4C),(c₊₁,c⁻¹)-sampling (FIG. 4D), (c₊₃, c⁻³)-sampling (FIG. 4E),(c₊₁,c⁻¹,c₊₃,c⁻³)-sampling (FIG. 4F), (c₊₀,c⁻²)-sampling (FIG. 4G),(c⁻⁰,c₊₂)-sampling (FIG. 4H), and (c₊₀,c⁻²,c⁻⁰,c₊₂)-sampling (FIG. 4I).Note that delayed acquisition for t₁(¹³C) greatly amplifies the salientfeatures of the various sampling schemes. All PMS and DPMS spectra wereobtained without a phase correction. The measurement time for the PMS 2D[¹³C,¹H]-HSQC spectra was one hour resulting in measurement time of twohours for the DMPS spectrum. The respective measurement times were twohours invested for the ‘States’ and dual ‘States’ spectra.

FIG. 5 illustrates cross sections along ω₁(¹³C) taken from 2D[¹³C^(aliphatic)/¹³C^(aromatic),¹H]-HSQC acquired with States (Ernst etal., “Principles of Nuclear Magnetic Resonance in One and TwoDimensions,” Oxford: Oxford University Press (1987), which is herebyincorporated by reference in its entirety), (c₊₁,c⁻¹)-PMS, or(c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS. The States spectrum was phase-corrected suchthat aliphatic peaks are purely absorptive. This leads to a dispersivecomponent in the aromatic peaks (top). The quad peak in the PMS spectra(middle) results from the dispersive component and is marked with (*).The quad peak is canceled in the clean absorption mode DPMS spectrum(bottom). The actual chemical shifts (detected without folding) areindicated. For data processing, see discussion in Example 1.

FIGS. 6A-B illustrate cross sections taken along ω₁(¹H) (FIG. 6A) andω₂(¹³C) (FIG. 6B) from 3D HC(C)H TOCSY spectra recorded with either‘States’ quadrature (Ernst et al., “Principles of Nuclear MagneticResonance in One and Two Dimensions,” Oxford: Oxford University Press(1987), which is hereby incorporated by reference in its entirety)detection or DPMS in both indirect dimensions. The latter yields a cleanabsorption mode spectrum without applying a phase correction. Note thatthe dispersive components of the peak located approximately in themiddle of the selected spectral range cannot be removed by a first orderphase correction: this would introduce dispersive components for otherpeaks located either up- or down-field. For data processing, seediscussion in Example 1.

FIG. 7 illustrates cross sections along ω₁(¹³C^(α);¹³CU¹³) taken fromthe ω₂(¹⁵N)-projection of the (4,3)D C ^(αβ) C ^(α)(CO)NHN (Atreya etal., Proc. Natl. Acad. Sci. USA, 101:9642-9647 (2004), which is herebyincorporated by reference in its entirety) sub-spectrum comprisingsignals at Ω(¹³C^(α))+Ω(¹³C^(αβ)), recorded with standard GFT NMR dataacquisition (Atreya et al., Proc. Natl. Acad. Sci. USA, 101:9642-9647(2004), which is hereby incorporated by reference in its entirety) or(c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS for the jointly sampled chemical shift evolutionperiods. The latter yields clean absorption mode GFT NMR sub-spectra.For data processing, see discussion in Example 1.

FIGS. 8A-D show cross sections taken along ω₁(¹³C) of aliphatic ct-2D[¹³C, ¹H]-HSQC spectra recorded with (c⁻¹,c⁻¹)-sampling (FIG. 8A),(c⁻³,c⁻³)-sampling (FIG. 8B), (c⁻⁰,c⁻²)-sampling (FIG. 8C), and(c⁻⁰,c₊₂)-sampling (FIG. 8D) for a 5 mM aqueous solution ofphenylalanine. The quadrature peaks for spectra recorded using(c⁻⁰,c⁻²)- and (c⁻⁰,c₊₂)-sampling (labeled with an asterisk) are shownafter a π/2 zero-order phase correction was applied. The digitalresolution after zero filling in the ψ₂(¹H) and ω₁(¹³C) dimensions are,respectively, 7.8 and 2.5 Hz/pt. The measurement time for each of thespectra was 40 minutes.

DETAILED DESCRIPTION OF THE INVENTION

The present invention relates to a method of conducting an N-dimensionalNMR experiment in a phase-sensitive manner by use of forward (from time0 to +t) and backward (from time 0 to −t) sampling of time domainshifted by a primary phase shift under conditions effective to measuretime domain amplitudes and secondary phase shifts. The method comprisesproviding a sample; applying radiofrequency pulses for an N-dimensionalNMR experiment to said sample; selecting m dimensions of said NMRexperiment, wherein m≦N, sampling a time domain modulation in aphase-sensitive manner in each selected dimension jε[1,2, . . . , m]arising from time evolution of chemical shift α_(j) in both a forwardand backward manner to obtain two interferograms for each time domaindimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},{wherein}}\end{matrix}$

I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j) ⁻are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j)are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[ and the cases{ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted,and Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻ are secondary phaseshifts; multiplying each said vectors C_(j,ψ) _(j) (t_(j)) with aD-matrix defined as

$D_{j} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}$

and a vector Q=[1 i], wherein i=√{square root over (−1)}, according toQ·D_(j)·C_(j,ψ) _(j) (t_(j)) under conditions effective to create acomplex time domain of said selected m dimensions according to

${\underset{j}{\otimes}{Q \cdot D_{j} \cdot {C_{j,\psi_{j}}\left( t_{j} \right)}}};$

and transforming said complex time domain into frequency domain by useof an operator O under conditions effective to measure the values ofI_(j,ψ) _(j) ⁺, I_(j,ψ) _(j) _(,δ) _(j) ⁻, Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ)_(j) _(,δ) _(j) ⁻ in said frequency domain.

Another aspect of the present invention relates to a method ofconducting an N-dimensional NMR experiment in a phase-sensitive mannerby use of dual forward (from time 0 to +t) and backward (from time 0 to−t) sampling of time domain shifted by a primary phase shift underconditions effective to measure secondary phase shifts or at leastpartially cancel dispersive and quadrature image signal componentsarising in a frequency domain from secondary phase shifts. The methodcomprises providing a sample; applying radiofrequency pulses for anN-dimensional NMR experiment to said sample; selecting m dimensions ofsaid NMR experiment, wherein m≦N, sampling a time domain modulation in aphase-sensitive manner in each said selected dimension jε[1,2, . . . ,m] arising from time evolution of chemical shift α_(j) in both a forwardand backward manner to obtain two interferograms for each time domaindimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},{wherein}}\end{matrix}$

I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j) _(,δ) _(j) ⁻ are amplitudes, Ψ_(j) andΨ_(j)+δ_(j) are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[and thecases {ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 beingomitted, and Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻ are secondaryphase shifts; multiplying each said vectors C_(j,ψ) _(j (t) _(j)) with aD-matrix defined as

$D_{j} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}$

and a vector Q=[1 i], wherein i=√{square root over (−1)}, according toQ·D_(j)·C_(j,ψ) _(j) (t_(j)) under conditions effective to create acomplex time domain of said selected m dimensions according to

${\underset{j}{\otimes}{Q \cdot D_{j} \cdot {C_{j,\psi_{j}}\left( t_{j} \right)}}};$

repeating said selecting, said sampling and said multiplying(2^(m))-times, thereby sampling the m dimensions with all 2^(m) possiblepermutations resulting from selecting for each dimension j either Ψ_(j)or Ψ_(j)+π/2, with δ_(j) being incremented by either 0 or π, therebyyielding 2^(m) complex time domains; linearly combining said 2^(m)complex time domains; and transforming said linearly combined complextime domain into frequency domain by use of an operator O, underconditions effective to at least partially cancel dispersive andquadrature image peak components arising from Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ)_(j) _(,δ) _(j) ⁻ in said frequency domain.

A further aspect of the present invention relates to a method ofconducting an N-dimensional NMR experiment in a phase-sensitive mannerby use of dual forward (from time 0 to +t) and backward (from time 0 to−t) sampling of time domain shifted by a primary phase shift underconditions effective to measure secondary phase shifts or at leastpartially cancel dispersive and quadrature image signal componentsarising in a frequency domain from secondary phase shifts. The methodcomprises providing a sample; applying radiofrequency pulses for anN-dimensional NMR experiment to said sample; selecting m dimensions ofsaid NMR experiment, wherein m≦N; sampling twice a time domainmodulation in a phase-sensitive manner in each said selected dimensionjε=[1,2, . . . , m] arising from time evolution of chemical shift α_(j),once in a forward manner to obtain two interferograms for each timedomain dimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}^{+}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{c_{\psi_{j},\delta_{j}}^{+}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{\cos \left( {\psi_{j} + \delta_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}}\end{bmatrix}},{and}}\end{matrix}$

once in a backward manner to obtain two interferograms for each timedomain dimension t_(j) defining the vector

$\begin{matrix}{{C_{j,\psi_{j}}^{-}\left( t_{j} \right)}:=\begin{bmatrix}{I_{j,\psi_{j}}^{-}{c_{\psi_{j}}^{-}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{-}{\cos \left( {\psi_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{-}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},}\end{matrix}$

wherein I_(j,ψ) _(j) ⁺, I_(j,ψ) _(j) _(,δ) _(j) ⁺, I_(j,ψ) _(j) ⁻, andI_(j,ψ) _(j) _(,δ) _(j) ⁻ are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j) areprimary phase shifts with Ψ_(j), δ_(j)ε[0,2π[, and Φ_(j,ψ) _(j) ⁺,Φ_(j,ψ) _(j) _(,δ) _(j) ⁺, Φ_(j,ψ) _(j) ⁻, and I_(j,ψ) _(j) _(,δ) _(j) ⁻are secondary phase shifts; multiplying each said vector C_(j,ψ) _(j) ⁺W(t_(j)) with a D-matrix defined as

$D_{j}^{+} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{\cos \left( {\psi_{j} + \delta_{j}} \right)} & {- {\cos \left( \psi_{j} \right)}}\end{bmatrix}$

and each said vector C_(j,ψ) _(j) ⁻(t_(j)) with a D-matrix defined as

${D_{j}^{-} = \begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}};$

multiplying the said products D_(j) ⁺·C_(j,ψ) _(j) ⁺(t_(j)) and D_(j)⁻·C_(j,ψ) _(j) ⁻(t_(j)) with a vector Q=[1 i], wherein i=√{square rootover (−1)}, according to Q·D_(j) ⁺·C_(j,ψ) _(j) ⁺(t_(j)) and Q·D_(j)⁻·C_(j,ψ) _(j) ⁻(t_(j)) under conditions effective to create a complextime domain of said selected m dimensions according to

${\underset{j}{\otimes}{{Q \cdot D_{j}^{+} \cdot {C_{j,\psi_{j}}^{+}\left( t_{j} \right)}}\mspace{14mu} {{{and}\mspace{14mu} \underset{j}{\otimes}Q} \cdot D_{j}^{-} \cdot {C_{j,\psi_{j}}^{-}\left( t_{j} \right)}}}};$

repeating said selecting, said phase-sensitive sampling twice and saidmultiplying (2^(m)−2)-times, thereby sampling the said m dimensions withall 2^(m) possible permutations resulting from selecting for eachdimension j either phase-sensitive forward or backward samplingaccording to C_(j,ψ) _(j) ⁺(t_(j)) or C_(j,ψ) _(j) ⁻(t_(j)); linearlycombining said 2^(m) complex time domains; and transforming saidlinearly combined complex time domain into frequency domain by use of anoperator O, under conditions effective to at least partially canceldispersive and quadrature image peak components arising from Φ_(j,ψ)_(j) ⁺, Φ_(j,ψ) _(j) _(,δ) _(j) ⁺, Φ_(j,ψ) _(j) ³¹ , and Φ_(j,ψ) _(j)_(,δ) _(j) in said frequency domain.

Suitable NMR experiments for the present invention include, but are notlimited to, those described in U.S. Pat. Nos. 6,831,459, 7,141,432,7,365,539, 7,396,685, and 7,408,346, which are hereby incorporated byreference in their entirety. Any desired sample suitable for NMRexperiments may be used.

In the present invention, novel, generally applicable acquisitionschemes for phase-sensitive detection of pure absorption mode signalsdevoid of dispersive components are described. They were established bygeneralizing mirrored time domain sampling (MS) to ‘phase shifted MS’(PMS). MS was originally contemplated for absolute-value 2D resolved NMRspectroscopy (Bachmann et al., J. Magn. Reson., 28:29-39 (1977), whichis hereby incorporated by reference in its entirety) and was laterintroduced for measurement of spin-spin couplings in J-GFT NMR (Atreyaet al., J. Am Chem. Soc. 129:680-692 (2007), which is herebyincorporated by reference in its entirety).

Phase-sensitive detection of a chemical shift a can be accomplished bysampling the time evolution (‘precession’) of transverse magnetizationtwice, under the condition that the two axes along which the timeevolution is sampled and thus registered, which yields a time domain‘interferogram’, are not collinear. When considering that the timedomain can be sampled from time 0 to t (‘forward sampling’), as well asfrom time 0 to −t (backward sampling), three cases can be considered:

-   -   (i) Both the interferograms are forward sampled from time 0 to        t,    -   (ii) Both the interferograms are backward sampled from time 0 to        −t and    -   (iii) Combined forward and backward sampling, where one of the        two interferograms is forward sampled from time 0 to −t and the        other is backward sampled from time 0 to −t (also referred to        herein as “mirrored time domain sampling”).

(i) Both Interferograms are Forward Sampled

The two interferograms for the j^(th) time dimension in amultidimensional NMR experiment are given by

$\begin{matrix}\begin{matrix}{{C_{j,\psi_{j}}^{+}\left( t_{j} \right)} = \begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{c_{\psi_{j},\delta_{j}}^{+}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{\cos \left( {\psi_{j} + \delta_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}}\end{bmatrix}},}\end{matrix} & ({B1})\end{matrix}$

where ψ_(j) is a primary phase shift, δ_(j) is the difference of primaryphase shifts of the two interferograms, I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j)_(,δ) _(j) ⁺ are the amplitudes of the modulations, and Φ_(j,ψ) _(j) ⁺,and Φ_(j,ψ) _(j) _(,δ) _(j) are defining secondary phase shifts.Orthogonal phase-sensitive forward sampling, requiring that δ_(j)=π/2,represents known art in the field and is commonly referred to as‘States’ sampling or ‘quadrature detection’ (States et al., J. Magn.Reson., 48:286-292 (1982), which is hereby incorporated by reference inits entirety.

The complex time domain signal S_(j,ψ) _(j) ⁺(t_(j)) resulting fromsignal detection along the two non-collinear axes is proportional to

$\begin{matrix}\begin{matrix}{{{S_{j,\psi_{j}}^{+}\left( t_{j} \right)} \propto {{QD}_{j}^{+}{C_{j,\psi_{j}}^{+}\left( t_{j} \right)}}} = {\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{\cos \left( {\psi_{j} + \delta_{j}} \right)} & {- {\cos \left( \psi_{j} \right)}}\end{bmatrix}}} \\{\begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{+}{c_{\psi_{j},\delta_{j}}^{+}\left( t_{j} \right)}}\end{bmatrix}} \\{= {\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{\cos \left( {\psi_{j} + \delta_{j}} \right)} & {- {\cos \left( \psi_{j} \right)}}\end{bmatrix}}} \\{\begin{bmatrix}\begin{matrix}I_{j,\psi_{j}}^{+} \\\begin{pmatrix}{{\cos \; \psi_{j}{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} -} \\{\sin \; \psi_{j}{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}}\end{pmatrix}\end{matrix} \\\begin{matrix}I_{j,\psi_{j},\delta_{j}}^{+} \\\begin{pmatrix}\begin{matrix}{\cos \; \left( {\psi_{j} + \delta_{j}} \right)} \\{{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)} -}\end{matrix} \\\begin{matrix}{\sin \; \left( {\psi_{j} + \delta_{j}} \right)} \\{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}\end{matrix}\end{pmatrix}\end{matrix}\end{bmatrix}} \\{= {{\sin \; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{+} \\{{\cos \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} -}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}^{+}} \\{{\sin \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}}^{+} \\{{\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}} +}\end{matrix}} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{+}} \\{\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}}\end{matrix}\end{pmatrix}}} -}} \\{{{\sin \; \psi_{j}\cos \; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{+} \\{{\sin \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}^{+}} \\{{\cos \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}}^{+} \\{{\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}} -}\end{matrix}} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{+}} \\{\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}}\end{matrix}\end{pmatrix}}} +}} \\{{{\cos^{2}\psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{+} \\{{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{+}} \\{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}\end{matrix}\end{pmatrix}}} +}} \\{{{\sin^{2}\psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j},\delta_{j}}^{+} \\{{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}^{+}} \\{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}\end{matrix}\end{pmatrix}}} -}} \\{{{\sin^{2}\psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{+} \\{{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}}^{+} \\{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}\end{matrix}}\end{pmatrix}}} +}} \\{{\cos^{2}\psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}{\; I_{j,\psi_{j}}^{+}} \\{{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)} -}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{+}} \\{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}\end{matrix}\end{pmatrix}}}}\end{matrix} & ({B2})\end{matrix}$

(ii) Both Interferograms are Backward Sampled

The two interferograms for the j^(th) time dimension in amultidimensional NMR experiment are given by

$\begin{matrix}\begin{matrix}{{C_{j,\psi_{j}}^{-}\left( t_{j} \right)} = \begin{bmatrix}{I_{j,\psi_{j}}^{-}{c_{\psi_{j}}^{-}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{-}{\cos \left( {\psi_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{-}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},}\end{matrix} & ({B3})\end{matrix}$

where ψ_(j) is a primary phase shift, δ_(j) is the difference of primaryphase shifts of the two interferograms, I_(j,ψ) _(j) ⁻ and I_(j,ψ) _(j)_(,δ) _(j) ⁻ are the amplitudes of the modulations, and Φ_(j,ψ) _(j) ⁻and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻ are defining secondary phase shifts.Orthogonal phase-sensitive backward sampling, requiring that δ_(j)=π/2,represents known art in the field and is also commonly referred to as‘States’ sampling or ‘quadrature detection’ (States et al., J. Magn.Reson., 48:286-292 (1982), which is hereby incorporated by reference inits entirety).

The complex time domain signal S_(j,ψ) _(j) ⁻ (t_(j)) resulting fromsignal detection along the two non-collinear axes is proportional to

$\begin{matrix}\begin{matrix}{{{S_{j,\psi_{j}}^{-}\left( t_{j} \right)} \propto {{QD}_{j}^{-}{C_{j,\psi_{j}}^{-}\left( t_{j} \right)}}} = {\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{I_{j,\psi_{j}}^{-}{c_{\psi_{j}}^{-}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{= {\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}}} \\{\begin{bmatrix}\begin{matrix}I_{j,\psi_{j}}^{-} \\\begin{pmatrix}{{\cos \; \psi_{j}{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)}} +} \\{\sin \; \psi_{j}{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)}}\end{pmatrix}\end{matrix} \\\begin{matrix}I_{j,\psi_{j},\delta_{j}}^{-} \\\begin{pmatrix}\begin{matrix}{\cos \; \left( {\psi_{j} + \delta_{j}} \right)} \\{{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)} +}\end{matrix} \\\begin{matrix}{\sin \; \left( {\psi_{j} + \delta_{j}} \right)} \\{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}\end{matrix}\end{pmatrix}\end{matrix}\end{bmatrix}} \\{= {{\sin \; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{-} \\{{\cos \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}^{-}} \\{{\sin \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}}^{-} \\{{\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}} -}\end{matrix}} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{-}} \\{\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}}\end{matrix}\end{pmatrix}}} -}} \\{{{\sin \; \psi_{j}\cos \; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{-} \\{{\sin \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} -}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}^{-}} \\{{\cos \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}}^{-} \\{{\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}} +}\end{matrix}} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{-}} \\{\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}}\end{matrix}\end{pmatrix}}} +}} \\{{{\cos^{2}\psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{-} \\{{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{-}} \\{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}\end{matrix}\end{pmatrix}}} +}} \\{{{\sin^{2}\psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j},\delta_{j}}^{-} \\{{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}^{-}} \\{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)}\end{matrix}\end{pmatrix}}} +}} \\{{{\sin^{2}\psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}}^{-} \\{{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}}^{-} \\{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}\end{matrix}}\end{pmatrix}}} -}} \\{{\cos^{2}\psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}{\; I_{j,\psi_{j}}^{-}} \\{{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)} -}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}^{-}} \\{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}\end{matrix}\end{pmatrix}}}}\end{matrix} & ({B4})\end{matrix}$

Addition of S_(j,ψ) _(j) ⁺(t_(j)) and S_(j,ψ) _(j) ⁻(t_(j)) then yieldsthe complex time domain signal for ‘dual States’ sampling and isproportional to

$\begin{matrix}{{{S_{j,\psi_{j}}^{+}\left( t_{j} \right)} + {S_{j,\psi_{j}}^{-}\left( t_{j} \right)}} \propto {{\sin \; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j}}^{+}\; \cos \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} -} \\{{\; I_{j,\psi_{j}}^{+}\sin \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} -} \\{{I_{j,\psi_{j},\delta_{j}}^{+}\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}} +} \\{{\; I_{j,\psi_{j},\delta_{j}}^{+}\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}} +} \\{{I_{j,\psi_{j}}^{-}\; \cos \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} +} \\{{\; I_{j,\psi_{j}}^{-}\sin \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} -} \\{{I_{j,\psi_{j},\delta_{j}}^{-}\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}} -} \\{\; I_{j,\psi_{j},\delta_{j}}^{-}\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}}\end{pmatrix}}} - {\sin \; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j}}^{+}\; \sin \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} +} \\{{\; I_{j,\psi_{j}}^{+}\cos \; \Phi_{j,\psi_{j}}^{+}^{{- {\alpha}}\; t}} -} \\{{I_{j,\psi_{j},\delta_{j}}^{+}\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}} -} \\{{\; I_{j,\psi_{j},\delta_{j}}^{+}\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{+}^{{- {\alpha}}\; t}} +} \\{{I_{j,\psi_{j}}^{-}\; \sin \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} -} \\{{\; I_{j,\psi_{j}}^{-}\cos \; \Phi_{j,\psi_{j}}^{-}^{{- {\alpha}}\; t}} -} \\{{I_{j,\psi_{j},\delta_{j}}^{-}\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}} +} \\{\; I_{j,\psi_{j},\delta_{j}}^{-}\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{-}^{{- {\alpha}}\; t}}\end{pmatrix}}} + {\cos^{2}\; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j}}^{+}\; {\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}^{+}} \right)}} +} \\{{\; I_{j,\psi_{j}}^{+}\sin \; \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)} +} \\{{I_{j,\psi_{j}}^{-}\cos \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)} +} \\{\; I_{j,\psi_{j},\delta_{j}}^{-}\sin \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}\end{pmatrix}}} + {\sin^{2}\; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j},\delta_{j}}^{+}\; {\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}} +} \\{{\; I_{j,\psi_{j}}^{+}\sin \; \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)} +} \\{{I_{j,\psi_{j},\delta_{j}}^{-}\cos \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)} +} \\{\; I_{j,\psi_{j}}^{-}\sin \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)}\end{pmatrix}}} - {\sin^{2}\; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j}}^{+}\; {\sin \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}^{+}} \right)}} -} \\{\; {{I_{j,\psi_{j},\delta_{j}}^{+}\sin \; \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)} -}} \\{{I_{j,\psi_{j}}^{-}\sin \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)} +} \\{\; {I_{j,\psi_{j},\delta_{j}}^{-}\sin \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{pmatrix}}} + {\cos^{2}\; \psi_{j}\cos \; {{\delta_{j}\begin{pmatrix}{{\; I_{j,\psi_{j}}^{+}\; {\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}^{+}} \right)}} -} \\{{\; I_{j,\psi_{j},\delta_{j}}^{+}\cos \; \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)} -} \\{{\; I_{j,\psi_{j}}^{-}\cos \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j}}^{-}} \right)} +} \\{\; I_{j,\psi_{j},\delta_{j}}^{-}\cos \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}\end{pmatrix}}.}}}} & ({B5})\end{matrix}$

For identical amplitudes, that is I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j) _(,δ) _(j)⁺=I_(j,ψ) _(j) ⁻=I_(j,ψ) _(j) _(,δ) _(j) ⁺=I_(j), and identicalsecondary phases, that is Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j) _(,δ) _(j)⁺=Φ_(j,ψ) _(j) ⁻=Φ_(j,ψ) _(j) _(,δ) _(j) ⁻Eq. (B5) simplifies to

S_(j,ψ) _(j) ⁺(t_(j))+S_(j,ψ) _(j) ⁻(t_(j))∝2 sin δ_(j)I_(j) cosΦ_(j)e^(iαt)  (B6).

Inspection of Eq. (B6) reveals that Fourier Transformation (FT) ofsignals acquired by use of ‘dual States’ yields absorptive spectrum.Furthermore, the intensity of the frequency domain peaks is maximum fororthogonal sampling (δ_(j)=π/2 or 3π/2).

(iii) One Interferogram is Forward and the Other Backward Sampled

The two interferograms for the j^(th) time dimension in amultidimensional NMR experiment are given by

$\begin{matrix}\begin{matrix}{{C_{j,\psi_{j}}\left( t_{j} \right)} = \begin{bmatrix}{I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}}\end{bmatrix}} \\{{= \begin{bmatrix}{I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\{I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}}\end{bmatrix}},}\end{matrix} & ({B7})\end{matrix}$

where ψ_(j) is a primary phase shift, δ_(j) is the difference of primaryphase shifts of the two interferograms, I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j)_(,δ) _(j) ⁻ are the amplitudes of the modulations, and Φ_(j,ψ) _(j) ⁻and Φ_(j,ψ) _(j) _(,δ) _(j) ⁻are defining secondary phase shifts.Phase-sensitive signal detection requires that ψ_(j)≠nπ/2 if δ_(j)=0.

The resulting complex time domain signal S_(j,ψ) _(j) (t_(j)) isproportional to

$\begin{matrix}\begin{matrix}{{{S_{j,\psi_{j}}\left( t_{j} \right)} \propto {{QD}_{j}{C_{j,\psi_{j}}\left( t_{j} \right)}}} = {\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{I_{j,\psi_{j}}{c_{\psi_{j}}\left( t_{j} \right)}} \\{I_{j,\psi_{j},\delta_{j}}{c_{\psi_{j},\delta_{j}}\left( t_{j} \right)}}\end{bmatrix}} \\{= {\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}{\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\{- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)}\end{bmatrix}}} \\{\begin{bmatrix}\begin{matrix}I_{j,\psi_{j}} \\\begin{pmatrix}{{\cos \; \psi_{j}{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}} \right)}} -} \\{\sin \; \psi_{j}{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}} \right)}}\end{pmatrix}\end{matrix} \\\begin{matrix}I_{j,\psi_{j},\delta_{j}} \\\begin{pmatrix}\begin{matrix}{\cos \; \left( {\psi_{j} + \delta_{j}} \right)} \\{{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}} \right)} +}\end{matrix} \\\begin{matrix}{\sin \; \left( {\psi_{j} + \delta_{j}} \right)} \\{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}} \right)}\end{matrix}\end{pmatrix}\end{matrix}\end{bmatrix}} \\{= {{\sin \; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}} \\{{\cos \; \Phi_{j,\psi_{j}}^{{- {\alpha}}\; t}} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}} \\{{\sin \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} +}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}} \\{{\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}} -}\end{matrix}} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}} \\{\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}}\end{matrix}\end{pmatrix}}} -}} \\{{{\sin \; \psi_{j}\cos \; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}} \\{{\sin \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} -}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}} \\{{\cos \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} +}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}} \\{{\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}} +}\end{matrix}} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}} \\{\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}}\end{matrix}\end{pmatrix}}} +}} \\{{{\cos^{2}\psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}} \\{{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}} \right)} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}} \\{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}} \right)}\end{matrix}\end{pmatrix}}} -}} \\{{{\sin^{2}\psi_{j}\sin \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j},\delta_{j}} \\{{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}} \right)} +}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j}}} \\{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}} \right)}\end{matrix}\end{pmatrix}}} -}} \\{{{\sin^{2}\psi_{j}\cos \; {\delta_{j}\begin{pmatrix}\begin{matrix}I_{j,\psi_{j}} \\{{\sin \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}} \right)} -}\end{matrix} \\{\; \begin{matrix}I_{j,\psi_{j},\delta_{j}} \\{\sin \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}} \right)}\end{matrix}}\end{pmatrix}}} -}} \\{{\cos^{2}\psi_{j}\cos \; {{\delta_{j}\begin{pmatrix}\begin{matrix}{\; I_{j,\psi_{j}}} \\{{\cos \left( {{\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}} \right)} -}\end{matrix} \\\begin{matrix}{\; I_{j,\psi_{j},\delta_{j}}} \\{\cos \left( {{\alpha_{j}t_{j}} - \Phi_{j,\psi_{j},\delta_{j}}} \right)}\end{matrix}\end{pmatrix}}.}}}\end{matrix} & ({B8})\end{matrix}$

When increasing ψ_(j) by π/2 while δ_(j) remains unchanged, S_(j,ψ) _(j)(t_(j)) takes the form

$\begin{matrix}{{S_{j,{\psi_{j} + \frac{\pi}{2}}}\left( t_{j} \right)} \propto {{{- \sin}\; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} +} \\{{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} +} \\{{I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}} -} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}}\end{pmatrix}}} + {\sin \; \psi_{j}\cos \; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} -} \\{{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} +} \\{{I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}} +} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}}\end{pmatrix}}} + {\cos^{2}\; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}}}{\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}} +} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\sin \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}\end{pmatrix}}} - {\sin^{2}\; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}{\cos \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}} +} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}\sin \; \left( {{\alpha_{j}t_{j}} + \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}\end{pmatrix}}} - {\sin^{2}\; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}}}{\sin \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}} -} \\{\; {I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\sin \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}}\end{pmatrix}}} - {\cos^{2}\; \psi_{j}\cos \; {{\delta_{j}\begin{pmatrix}{{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}{\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}} -} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\cos \; \left( {{\alpha_{j}t_{j}} - \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}\end{pmatrix}}.}}}} & ({B9})\end{matrix}$

Subtraction of

$S_{j,{\psi_{j} + \frac{\pi}{2}}}\left( t_{j} \right)$

from S_(j,ψ) _(j (t) _(j)) yields the complex time domain signal forsuch ‘dual combined forward and backward’ sampling, which isproportional to

$\begin{matrix}{{{S_{j,\psi_{j}}\left( t_{j} \right)} + {S_{j,{\psi_{j} + \frac{\pi}{2}}}\left( t_{j} \right)}} \propto {{\sin \; \psi_{j}\cos \; \psi_{j}\cos \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j}}\cos \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} +} \\{{\; I_{j,\psi_{j}}\sin \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} +} \\{{I_{j,\psi_{j},\delta_{j}}\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}} -} \\{{\; I_{j,\psi_{j},\delta_{j}}\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}} +} \\{{I_{j,{\psi_{j} + \frac{\pi}{2}}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} +} \\{{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} +} \\{{I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}} -} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}}\end{pmatrix}}} - {\sin \; \psi_{j}\cos \; \psi_{j}\sin \; {\delta_{j}\begin{pmatrix}{{I_{j,\psi_{j}}\sin \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} -} \\{{\; I_{j,\psi_{j}}\cos \; \Phi_{j,\psi_{j}}^{{\alpha}\; t}} +} \\{{I_{j,\psi_{j},\delta_{j}}\sin \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}} +} \\{{\; I_{j,\psi_{j},\delta_{j}}\cos \; \Phi_{j,\psi_{j},\delta_{j}}^{{\alpha}\; t}} +} \\{{I_{j,{\psi_{j} + \frac{\pi}{2}}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} -} \\{{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}^{{\alpha}\; t}} +} \\{{I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\sin \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}} +} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}\cos \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{{\alpha}\; t}}\end{pmatrix}}} + {\sin \; {\delta_{j}\begin{pmatrix}{{\cos^{2}{\psi_{j}\begin{pmatrix}{{I_{j,\psi_{j}}{\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}} \right)}} +} \\{\; I_{j,\psi_{j},\delta_{j}}{\sin \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,\psi_{j},\delta_{j}}} \right)}}\end{pmatrix}}} -} \\{\sin^{2}{\psi_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}}}{\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}} +} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}{\sin \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}}\end{pmatrix}}}\end{pmatrix}}} - {\sin \; {\delta_{j}\begin{pmatrix}{{\sin^{2}{\psi_{j}\begin{pmatrix}{{I_{j,\psi_{j}}^{-}{\cos \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,\psi_{j},\delta_{j}}} \right)}} +} \\{\; I_{j,\psi_{j}}{\sin \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}} \right)}}\end{pmatrix}}} -} \\{\cos^{2}{\psi_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}^{-}{\cos \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}} +} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}{\sin \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}}\end{pmatrix}}}\end{pmatrix}}} - {\cos \; {\delta_{j}\begin{pmatrix}{{\sin^{2}{\psi_{j}\begin{pmatrix}{{I_{j,\psi_{j}}{\sin \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}} \right)}} -} \\{I_{j,\psi_{j},\delta_{j}}{\sin \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,\psi_{j},\delta_{j}}} \right)}}\end{pmatrix}}} -} \\{\cos^{2}{\psi_{j}\begin{pmatrix}{{I_{j,{\psi_{j} + \frac{\pi}{2}}}{\sin \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}} -} \\{\; {I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}{\sin \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}}}\end{pmatrix}}}\end{pmatrix}}} - {\cos \; {{\delta_{j}\begin{pmatrix}{{\cos^{2}{\psi_{j}\begin{pmatrix}{{\; I_{j,\psi_{j}}{\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,\psi_{j}}} \right)}} -} \\{\; I_{j,\psi_{j},\delta_{j}}{\cos \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,\psi_{j},\delta_{j}}} \right)}}\end{pmatrix}}} -} \\{\sin^{2}{\psi_{j}\begin{pmatrix}{{\; I_{j,{\psi_{j} + \frac{\pi}{2}}}{\cos \left( {{\alpha_{j}t_{j}} + \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \right)}} -} \\{\; I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}{\cos \left( {{\alpha_{j}t_{j}} - \; \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \right)}}\end{pmatrix}}}\end{pmatrix}}.}}}} & ({B10})\end{matrix}$

For identical amplitudes, that is

$\begin{matrix}{I_{j,\psi_{j}} = I_{j,\psi_{j},\delta_{j}}} \\{= I_{j,{\psi_{j} + \frac{\pi}{2}}}} \\{= I_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \\{{= I_{j}},{and}}\end{matrix}$

identical secondary phases, that is

$\begin{matrix}{\Phi_{j,\psi_{j}} = \Phi_{j,\psi_{j},\delta_{j}}} \\{= \Phi_{j,{\psi_{j} + \frac{\pi}{2}}}} \\{= \Phi_{j,{\psi_{j} + \frac{\pi}{2}},\delta_{j}}} \\{{= \Phi_{j}},}\end{matrix}$

Eq. (B10) simplifies to

$\begin{matrix}{{{S_{j,\psi_{j}}\left( t_{j} \right)} - {S_{j,{\psi_{j} + \frac{\pi}{2}}}\left( t_{j} \right)}} \propto {2I_{j}{\sin \left( {{2\psi_{j}} + \delta_{j} + \Phi_{j}} \right)}{^{{\alpha}\; t}.}}} & ({B11})\end{matrix}$

Inspection of Eq. (B11) reveals that FT of dual combined forward andbackward sampling yields absorption mode spectra for|2ψ_(j)+δ_(j)+Φ_(j)|=(2n+1)*π/2.

Alternatively, the time domain signal of Eq, (B9) can also be obtainedwith an increasing of ψ_(j) by 3π/2 and an unchanged δ_(j), followed byaddition of the resulting signal with S_(j,ψ) _(j) (t_(j)).

In the following, explicit calculations are performed for primary phaseshifts which are integral multiples of π/4, that is ψ_(j)=nπ/4 andψ_(j)+δ_(j)=mπ/4 along with I_(j,ψ) _(j) =I_(j,ψ) _(j) _(,δ) _(j) =1, sothat, for brevity, interferograms corresponding to the two integers(n,m) are denoted as

$\begin{matrix}\begin{matrix}{{C_{{\pm n},{\pm m}}(t)} = \begin{bmatrix}{c_{\pm n}(t)} \\{c_{\pm m}(t)}\end{bmatrix}} \\{{= \begin{bmatrix}{\cos \left( {{{\pm \alpha}\; t} + \frac{n\; \pi}{4} + \Phi_{\pm n}} \right)} \\{\cos \left( {{{\pm \alpha}\; t} + \frac{m\; \pi}{4} + \Phi_{\pm m}} \right)}\end{bmatrix}},}\end{matrix} & ({B12})\end{matrix}$

where ‘±’ indicate forward (‘+’) and backward (‘−’) sampling. Table 1provides a survey of the cases discussed in the following.

TABLE 1 Definition of interferograms ψ_(j) δ_(j) n m Identities 0 π/2 02 $\quad\begin{matrix}{{c_{0}^{+}\left( t_{j} \right)} \equiv {c_{+ 0}\left( t_{j} \right)}} \\{{c_{0,\frac{\pi}{2}}^{+}\left( t_{j} \right)} \equiv {c_{+ 2}\left( t_{j} \right)}}\end{matrix}$ 0 π/2 0 2 $\quad\begin{matrix}{{c_{0}^{-}\left( t_{j} \right)} \equiv {c_{- 0}\left( t_{j} \right)}} \\{{c_{0,\frac{\pi}{2}}^{-}\left( t_{j} \right)} \equiv {c_{- 2}\left( t_{j} \right)}}\end{matrix}$ π/4 0 1 1 $\quad\begin{matrix}{{c_{\frac{\pi}{4}}^{+}\left( t_{j} \right)} \equiv {c_{+ 1}\left( t_{j} \right)}} \\{{c_{\frac{\pi}{4},0}^{-}\left( t_{j} \right)} \equiv {c_{- 1}\left( t_{j} \right)}}\end{matrix}$ 3π/4 0 3 3 $\quad\begin{matrix}{{c_{\frac{3\pi}{4}}^{+}\left( t_{j} \right)} \equiv {c_{+ 3}\left( t_{j} \right)}} \\{{c_{\frac{3\pi}{4},0}^{-}\left( t_{j} \right)} \equiv {c_{- 3}\left( t_{j} \right)}}\end{matrix}$ 0 π/2 0 2 $\quad\begin{matrix}{{c_{0}^{+}\left( t_{j} \right)} \equiv {c_{+ 0}\left( t_{j} \right)}} \\{{c_{0,\frac{\pi}{2}}^{-}\left( t_{j} \right)} \equiv {c_{- 2}\left( t_{j} \right)}}\end{matrix}$ π/2 3π/2 2 0 $\quad\begin{matrix}{{c_{\frac{\pi}{2}}^{+}\left( t_{j} \right)} \equiv {c_{+ 2}\left( t_{j} \right)}} \\{{c_{\frac{\pi}{2},\frac{3\; \pi}{2}}^{-}\left( t_{j} \right)} \equiv {c_{- 0}\left( t_{j} \right)}}\end{matrix}$

Equations are thus derived for the complex time domain signal S(t), fora given chemical shift a assuming that the two secondary phase shiftsare identical, that is φ_(±n)=φ_(±m)=φ.

Forward sampling with n=0 and 2 results in ‘States’ quadraturedetection, (Ernst et al., “Principles of Nuclear Magnetic Resonance inOne and Two Dimensions,” Oxford: Oxford University Press (1987);Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego:Academic Press (2007); Schmidt-Rohr et al., “MultidimensionalSolid-State NMR and Polymers,” New York: Academic Press (1994), whichare hereby incorporated by reference in their entirety) which is denoted(c⁻⁰,c₊₂)-sampling here and yields a signal S(t)∝cos Φe^(i+t)+sinΦe^(iπ/2)e^(iαt). Corresponding backward (c⁻⁰,c⁻²)-sampling yieldsS(t)∝cos Φe^(iαt)−sin Φe^(iπ/2)e^(iαt), so that addition of the twospectra (corresponding to ‘Dual States’ (c₊₀,c₊₂,c⁻⁰,c⁻²)-sampling)cancels the dispersive components. Thus, a clean absorption mode signalS(t)∝cos φ e^(iαt) is detected (FIGS. 1A-B).

Forward sampling and backward sampling with n=1 results in(c₊₁,c⁻¹)-PMS, which yields S(t)∝cos Φe^(iαt)−sin φe^(−iαt), i.e., twoabsorptive signals are detected: the desired signal at frequency α withrelative intensity cos Φ, and a quadrature image (‘quad’) peak atfrequency −α with intensity sin Φ. (c₊₁,c⁻¹)-PMS thus eliminates adispersive component by transformation into an absorptive quad peak.Without phase correction, this results in clean absorption mode signals(FIGS. 1A-B). Corresponding (c₊₃,c⁻³)-PMS sampling yields S(t)∝cosΦe^(iαt)+sin Φe^(−iαt), so that the quad peak is of opposite sign whencompared with (c₊₁,c⁻¹)-PMS. Addition of the two spectra cancels thequad peak (FIGS. 1A-B), and such combined (c₊₁,c⁻¹,c₊₃,c⁻³)-sampling isnamed dual PMS (DPMS).

Forward sampling with n=0 and backward sampling with n=2 results in(c⁻⁰,c⁻²)-PMS yielding S(t)∝cos Φe^(iαt)−sin Φe^(iπ/2)e^(−iαt), i.e.,the quad peak is dispersive. This feature allows one to distinguishgenuine and quad peaks if required. In (c⁻⁰,c₊₂)-PMS, the quad peak isof opposite sign when compared with (C₊₀,c⁻²)-PMS, i.e., S(t)∝cosΦe^(iαt)+sin Φe^(iπ/2)e^(−iαt). Thus, (c₊₀,c⁻²,c⁻⁰,c₊₂)-DPMS likewiseenables cancellation of the quad peak yielding solely clean absorptionmode signals.

PMS can be applied to an arbitrary number of indirect dimensions of amulti-dimensional experiment. For example, (c₊₁,c⁻¹)-PMS of K+1 chemicalshifts α₀, α₁, . . . α_(K) with phase errors Φ₀, Φ₁, . . . Φ_(K) yieldsa purely absorptive peak at (α₀, α₁, . . . α_(K)) with relativeintensity π^(K) _(j=0) cosΦ_(j), while the quad peak intensities areproportional to cos Φ_(j) for every +α_(j) and to sin Φ_(j) for every−α_(j) in the peak coordinates. PMS can likewise be applied to anarbitrary sub-set of the chemical shift evolution periods jointlysampled in GFT NMR. For example, joint (c₊₁,c⁻¹)-PMS of K+1 chemicalshifts α₀,α₁, . . . α_(K) yields a peak at the desired linearcombination of chemical shifts with relative intensity of π^(K) _(j=0)cos Φ_(j), while peaks located at different linear combinations ofshifts exhibit intensities proportional to cos Φ_(j) for all α_(j) forwhich the sign of the chemical shift in the linear combination does notchange, and proportional to sin Φ_(j) for all α_(j) for which the signin the linear combination does change. Hence, PMS converts dispersiveGFT NMR peak components into both quad and ‘cross-talk’ peaks. For agiven sub-spectrum, the latter peaks are located at linear combinationsof chemical shifts which are detected in the other sub-spectra (Kim etal., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl.Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579(2005); Yang et al., J. Am Chem. Soc. 127:9085-9099 (2005); Atreya etal., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad.Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am Chem. Soc.129:680-692 (2007), which are hereby incorporated by reference in theirentirety). Furthermore, arbitrary combinations of time domain samplingschemes can be employed in multi-dimensional NMR, including GFT NMR.

Clean absorption mode data acquisition leads to a reduction of thesignal maximum (and therefore the signal-to-noise ratio (S/N)) relativeto a hypothetical absorptive signal by a factor of cos Φ (see above). Itis therefore advantageous to employ the commonly used repertoire (Ernstet al., “Principles of Nuclear Magnetic Resonance in One and TwoDimensions,” Oxford: Oxford University Press (1987); Jacobsen, N. E.,“NMR Spectroscopy Explained,” Wiley, New York (2007); Cavanagh et al.,“Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007);Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,”New York: Academic Press (1994); Messerle et al., J. Magn. Reson.85:608-613 (1989); Keeler et al., Methods Enzymol. 239:145-207 (1994);Sorensen et al., J. Magn. Reson. 56:527-534 (1984), which are herebyincorporated by reference in their entirety) of techniques to avoidphase corrections, so that only residual dispersive components have tobe removed. For routine applications, however, the reduction in S/N isthen hardly significant: assuming that residual phase errors are|Φ|<15°, one obtains a reduction of <3.4%. Moreover, the superpositionof a dispersive component on an absorptive peak in a conventionallyacquired spectrum likewise reduces the signal maximum. As a result, theactual loss for |Φ|<15° is <1.7% (FIG. 2).

(c₊₁,c⁻¹)-PMS and (c₊₃,c⁻³)-PMS are unique since they yield cleanabsorption mode spectra (FIG. 1) with the same measurement time as isrequired for ‘States’ acquisition. Whenever the quad peaks (and crosstalk peaks in GFT NMR), which exhibit a relative intensity proportionalto sin Φ, emerge in otherwise empty spectral regions, they evidently donot interfere with spectral analysis and there is no need for theirremoval (when in doubt, (c₊₀,c⁻²)-PMS and (c⁻⁰,c₊₂)-PMS allows one toidentify quad peaks since they are purely dispersive). Furthermore,sensitivity limited data acquisition (Szyperski et al., Proc. Natl.Acad. Sci. USA 99:8009-8014 (2002), which is hereby incorporated byreference in its entirety) is often desirable (e.g., with an average S/N5). For |Φ|<15°, sin Φ<0.26 implies that quad and cross-talk peaksexhibit intensities ˜1.25 times the noise level, so that they are withinthe noise.

Suppression of axial peaks and residual solvent peaks is routinelyaccomplished using a two-step phase cycle (Ernst et al., “Principles ofNuclear Magnetic Resonance in One and Two Dimensions,” Oxford: OxfordUniversity Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,”2nd Ed., San Diego: Academic Press (2007), which are hereby incorporatedby reference in their entirety). In particular when studying moleculeswhich exhibit resonances close those of the solvent line (e.g. ¹H^(α)resonances of proteins dissolved in ¹H₂O) such additional suppression ofthe solvent line is most often required. DPMS schemes can be readilyconcatenated with the two-step cycle (see discussion below), that is,DPMS spectra can be acquired with the same measurement as a conventional2-step phase cycled NMR experiment. In solid state NMR relying on magicangle spinning of the sample (Schmidt-Rohr et al., “MultidimensionalSolid-State NMR and Polymers,” New York: Academic Press (1994), which ishereby incorporated by reference in its entirety), artifact suppressionrelies primarily on phase cycling, and such concatenation of (multiple)DPMS and phase cycles enables one to obtain clean absorption modespectra without investment of additional spectrometer time.

Application for Non-Identical Secondary Phase Shifts

One aspect of the present invention relates to a general application fornon-identical secondary phase shifts, where secondary phase shiftassociated with each interferogram is different from one another.

FT of S(t) yields, in general a frequency domain peak located at +αincluding an absorptive component and a dispersive component, and itsfrequency domain quadrature image peak (‘quad peak’) located at −α, alsoincluding an absorptive component and a dispersive component. These fourpeak components can be written as components of a vector F. With

A+ denoting the absorptive component of the peak at +α,D+ denoting the dispersive component of the peak at +α,A− denoting the absorptive component of the peak at −α, andD− denoting the dispersive component of the peak at −α,the frequency domain signal is then proportional to

$\begin{matrix}{{{{Re}\left( {F_{C}\left\lbrack {S(t)} \right\rbrack} \right)} \propto {\lambda F}} = {{\begin{bmatrix}\lambda^{A +} & \lambda^{D +} & \lambda^{A -} & \lambda^{D -}\end{bmatrix}\begin{bmatrix}{A +} \\{D +} \\{A -} \\{D -}\end{bmatrix}}.}} & (1)\end{matrix}$

F_(C) denotes the complex FT and λ=[λ^(A+)λ^(D+)λ^(A−)λ^(D−)] representsa ‘coefficient vector’ which provides the relative intensities ofabsorptive (A+) and dispersive (D+) components located at frequency +α,as well as the relative intensities of absorptive (A−) and dispersive(D−) components located at the quadrature image frequency −α.

Given the four coefficient vector components in Eq. 1, three figures ofmerit are defined to compare the sampling schemes:

$M^{D} = \frac{\lambda^{A +}}{{\lambda^{A +}} + {\lambda^{D +}}}$

1. Figure of Merit for elimination of the dispersive component,

2. Figure of Merit for elimination of the quadrature image,

$M^{Q} = \frac{{\lambda^{A +}} + {\lambda^{D +}}}{{\lambda^{A +}} + {\lambda^{D +}} + {\lambda^{A -}} + {\lambda^{D -}}}$

3. Figure of Merit for maximizing the intensity of the absorptivecomponent,

$M^{A} = \frac{\lambda^{A +}}{{\lambda^{A +}\left( {\Phi_{\pm n} = 0} \right)}}$

‘States’ Sampling

(c₊₀,c₊₂)-Sampling

The two interferograms acquired for ‘States’ quadrature detection (Ernstet al., “Principles of Nuclear Magnetic Resonance in One and TwoDimensions,” Oxford: Oxford University Press (1987); Jacobsen, N. E.,“NMR Spectroscopy Explained,” Wiley, New York (2007), which are herebyincorporated by reference in their entirety) are given by

$\begin{matrix}\begin{matrix}{{C_{{+ 0},{+ 2}}(t)} = \begin{bmatrix}{c_{+ 0}(t)} \\{c_{+ 2}(t)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \Phi_{+ 0}} \right)} \\{\cos \left( {{{+ \alpha}\; t} + \frac{\pi}{2} + \Phi_{+ 2}} \right)}\end{bmatrix}} \\{= {\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \Phi_{+ 0}} \right)} \\{- {\sin \left( {{{+ \alpha}\; t} + \Phi_{+ 2}} \right)}}\end{bmatrix}.}}\end{matrix} & (2)\end{matrix}$

The resulting complex time domain signal S_(+0,+2)(t) is proportional to

$\begin{matrix}{{{S_{{+ 0},{+ 2}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 0},{+ 2}} {C_{{+ 0},{+ 2}}( t)}}} = {{{\begin{bmatrix}1 & \end{bmatrix}\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}{c_{+ 0}(t)} \\{c_{+ 2}(t)}\end{matrix} \right\rbrack} = {\quad{{\begin{bmatrix}1 & {- }\end{bmatrix}\left\lbrack \begin{matrix}{\cos \left( {{{+ \alpha}\; t} + \Phi_{+ 0}} \right)} \\{- {\sin \left( {{{+ \alpha}\; t} + \Phi_{+ 2}} \right)}}\end{matrix} \right\rbrack} = {{\left( {{\cos \; \Phi_{+ 0}{\cos \left( {\alpha \; t} \right)}} - {\sin \; \Phi_{+ 0}{\sin \left( {\alpha \; t} \right)}}} \right) + {\left( {{\cos \; \Phi_{+ 2}{\sin \left( {\alpha \; t} \right)}} + {\sin \; \Phi_{+ 2}{\cos \left( {\alpha \; t} \right)}}} \right)}} = {{{\left( {{\cos \; \Phi_{+ 0}} + {\; \sin \; \Phi_{+ 2}}} \right){\cos \left( {\alpha \; t} \right)}} - {\left( {{\sin \; \Phi_{+ 0}} - {cos\Phi}_{+ 2}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\left( {{\cos \; \Phi_{+ 0}} + {\; \sin \; \Phi_{+ 2}}} \right)\frac{^{\; \alpha \; t} + ^{{- {\alpha}}\; t}}{2}} - {\left( {{\sin \; \Phi_{+ 0}} - {\; \cos \; \Phi_{+ 2}}} \right)\frac{^{\; \alpha \; t} - ^{{- {\alpha}}\; t}}{2}}} = {\quad{{{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}} \right)^{\; \alpha \; t}} + {\frac{}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}}} \right)^{{- }\; \alpha \; t}} - \mspace{185mu} {\frac{}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}}} \right) ^{{- }\; \alpha \; t}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}} \right) ^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}}} \right) ^{\; \frac{\pi}{2}} ^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}}} \right) ^{{- {\alpha}}\; t}} - \mspace{445mu} {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}}} \right)^{\; \frac{\pi}{2\;}}{^{{- }\; \alpha \; t}.}}}}}}}}}}}} & (3)\end{matrix}$

After FT, one obtains for (c₊₀,c₊₂)-sampling:

$\begin{matrix}{{{\lambda_{{+ 0},{+ 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}} \right)}};{\lambda_{{+ 0},{+ 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}}} \right)}}}{{\lambda_{{+ 0},{+ 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}}} \right)}};{\lambda_{{+ 0},{+ 2}}^{D -} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}}} \right)}}}{{{or}\mspace{14mu} {equivalently}},{\Phi_{+ 0} = {{\arccos \left( {\lambda_{{+ 0},{+ 2}}^{A +} + \lambda_{{+ 0},{+ 2}}^{A -}} \right)} = {\arcsin \left( {\lambda_{{+ 0},{+ 2}}^{D +} - \lambda_{{+ 0},{+ 2}}^{D -}} \right)}}}}{\Phi_{+ 2} = {{\arccos \left( {\lambda_{{+ 0},{+ 2}}^{A +} - \lambda_{{+ 0},{+ 2}}^{A -}} \right)} = {{\arcsin \left( {\lambda_{{+ 0},{+ 2}}^{D +} + \lambda_{{+ 0},{+ 2}}^{D -}} \right)}.}}}} & (4)\end{matrix}$

yielding superposition of absorptive and dispersive components at boththe actual and the quadrature peak positions. These equations enable oneto calculate the secondary phase shifts from the experimentally observedvalues for λ. The figures of merit for (c₊₀,c₊₂)-sampling are then givenby

$\begin{matrix}{M_{{+ 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}}}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}}} + {{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}}}}}} & (5) \\{M_{{+ 0},{+ 2}}^{Q} = \frac{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}}} + {{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}}} + {{{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}}}}{{{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}}}}} +} \\{{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}}}}\end{matrix}}} & \; \\{M_{{+ 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}}}}.}}} & \;\end{matrix}$

(c⁻⁰,c⁻²)-Sampling

For backward sampling the interferograms are given by

$\begin{matrix}\begin{matrix}{{C_{{- 0},{- 2}}(t)} = \begin{bmatrix}{c_{- 0}(t)} \\{c_{- 2}(t)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{- \alpha}\; t} + \Phi_{- 0}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{2} + \Phi_{{- 2}\;}} \right)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{- \alpha}\; t} + \Phi_{- 0}} \right)} \\{- {\sin \left( {{{- \alpha}\; t} + \Phi_{- 2}} \right)}}\end{bmatrix}} \\{= {\begin{bmatrix}{\cos \left( {{\alpha \; t} - \Phi_{- 0}} \right)} \\{\sin \left( {{\alpha \; t} - \Phi_{- 2}} \right)}\end{bmatrix}.}}\end{matrix} & (6)\end{matrix}$

so that the resulting signal s_(−0,−2)(t) is proportional to

$\begin{matrix}{{{S_{{- 0},{- 2}}( t)} \propto {\begin{bmatrix}1 & \end{bmatrix} D_{{- 0},{- 2}} {C_{{- 0},{- 2}}( t)}}} = {\quad{{{{{\begin{bmatrix}1 & \end{bmatrix}\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}{c_{- 0}(t)} \\{c_{- 2}(t)}\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}1 & \end{matrix} \right\rbrack}\left\lbrack \begin{matrix}c_{- 0} \\c_{- 2}\end{matrix} \right\rbrack} = {\quad{{\begin{bmatrix}1 & {- }\end{bmatrix}\begin{bmatrix}{\cos \left( {{\alpha \; t} - \Phi_{- 0}} \right)} \\{\sin \left( {{\alpha \; t} - \Phi_{- 2}} \right)}\end{bmatrix}} = {{\left( {{\cos \; \Phi_{- 0}{\cos \left( {\alpha \; t} \right)}} + {\sin \; \Phi_{- 0}{\sin \left( {\alpha \; t} \right)}}} \right) - {\left( {{\sin \; \Phi_{- 2}{\cos \left( {\alpha \; t} \right)}} - {\cos \; \Phi_{- 2}{\sin \left( {\alpha \; t} \right)}}} \right)}} = {{{\left( {{\cos \; \Phi_{- 0}} - {\; \sin \; \Phi_{- 2}}} \right){\cos \left( {\alpha \; t} \right)}} + {\left( {{\sin \; \Phi_{- 0}} + {cos\Phi}_{- 2}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\left( {{\cos \; \Phi_{- 0}} - {\; \sin \; \Phi_{- 2}}} \right)\frac{^{\; \alpha \; t} + ^{{- {\alpha}}\; t}}{2}} + {\left( {{\sin \; \Phi_{- 0}} - {\; \cos \; \Phi_{- 2}}} \right) \frac{^{\; \alpha \; t} - ^{{- {\alpha}}\; t}}{2}}} = {{{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} - {\frac{}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} + {\frac{}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right) ^{{- }\; \alpha \; t}}} = \mspace{160mu} {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right) ^{\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right) ^{\; \frac{\pi}{2}} ^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right) ^{{- {\alpha}}\; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)^{\; \frac{\pi}{2\;}}{^{{- }\; \alpha \; t}.}}}}}}}}}}}} & (7)\end{matrix}$

After FT, one obtains for (c⁻⁰,c⁻²)-sampling:

${\lambda_{{- 0},{- 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)}};$$\lambda_{{- 0},{- 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)}$${\lambda_{{- 0},{- 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)}};$$\lambda_{{- 0},{- 2}}^{D -} = {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)}$or  equivalently, Φ⁻⁰ = arccos (λ_(−0, −2)^(A+) + λ_(−0, −2)^(A−)) = −arcsin (λ_(−0, −2)^(D+) − λ_(−0, −2)^(D−))Φ⁻² = arccos (λ_(−0, −2)^(A+) − λ_(−0, −2)^(A−)) = −arcsin (λ_(−0, −2)^(D+) + λ_(−0, −2)^(D−)).

so that the figures of merit are given by

$\begin{matrix}{M_{{- 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}}{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}} + {{{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}}}}} & (9) \\{M_{{- 0},{- 2}}^{Q} = \frac{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}} + {{{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}} + {{{{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}}}{{{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}}}} +} \\{{{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}}}\end{matrix}}} & \; \\{M_{{- 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}}.}}} & \;\end{matrix}$

(c₊₀,c₊₂,c⁻⁰,c⁻²)-Sampling

Addition of S_(+0,+2)(t) and S_(−0,−2)(t) yields for such ‘dual Statessampling’

$\begin{matrix}{{S_{{+ 0},{+ 2},{- 0},{- 2}}(t)} = {{{S_{{+ 0},{+ 2}}(t)} + {S_{{- 0},{- 2}}(t)}} \propto {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)^{\; \frac{\pi}{2}}^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)^{^{\frac{\pi}{2}}}{^{{- }\; \alpha \; t}.}}}}} & (10)\end{matrix}$

After FT, one obtains for (c₊₀,c₊₂,c⁻⁰,c⁻²)-sampling:

$\begin{matrix}{{{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)}};}{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)}}{{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)}};}{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{D -} = {{- \frac{1}{2}}{\left( {{\cos \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right).}}}} & (11)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{+ 2},{- 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}} +} \\{{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}}}\end{matrix}}}{M_{{+ 0},{+ 2},{- 0},{- 2}}^{Q} = \frac{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}} +} \\{{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}}}\end{matrix}}{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}} +} \\{{{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}}} +} \\{{{{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}}} +} \\{{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}}}\end{matrix}}}M_{{+ 0},{+ 2},{- 0},{- 2}}^{A} = {\frac{1}{4}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}}}.}}} & (12)\end{matrix}$

π/4 and 3π/4-Shifted Mirrored Sampling

(c₊₁,c⁻¹)-Sampling (PMS)

The two interferograms for (c₊₁,c⁻¹)-PMS are given by

$\begin{matrix}\begin{matrix}{{C_{{+ 1},{- 1}}(t)} = \begin{bmatrix}{c_{+ 1}(t)} \\{c_{- 1}(t)}\end{bmatrix}} \\{= {\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 1}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 1}} \right)}\end{bmatrix}.}}\end{matrix} & (13)\end{matrix}$

so that the resulting signal S_(+1, −1)(t) is proportional to

$\begin{matrix}{{{S_{{+ 1},{- 1}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 1},{- 1}}{C_{{+ 1},{- 1}}(t)}}} = {{{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 1}(t)} \\{c_{- 1}(t)}\end{bmatrix}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}{1 - } & {1 + }\end{bmatrix}}\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 1}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 1}} \right)}\end{bmatrix}} = {{{\frac{1}{\sqrt{2}}\left( {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} + {\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right) - {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} - {\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right)}} \right){\cos \left( {\alpha \; t} \right)}} - {\frac{1}{\sqrt{2}}\left( {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} - {\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right) - {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} + {\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right)}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\frac{1}{\sqrt{2}}\left( {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} + {\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right) - {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} - {\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right)}} \right)\frac{^{{\alpha}\; t} + ^{{- }\; \alpha \; t}}{2}} - {\frac{1}{\sqrt{2}}\left( {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} - {\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right) - {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} + {\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}} \right)}} \right)\frac{^{\; \alpha \; t} - ^{{- {\alpha}}\; t}}{2}}} = {{{\frac{1}{4}\left( {{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}} + {\cos \; \Phi_{+ 1}} + {\sin \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}}} \right)^{\; \alpha \; t}} + {\frac{1}{4}\left( {{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}} - {\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}\sin \; \Phi_{+ 1}} -} \right)^{{- }\; \alpha \; t}} - {\frac{\;}{4}\left( {{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}} - {\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}}} \right)^{\; \alpha \; t}} - {\frac{\;}{4}\left( {{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}}} \right)^{{- }\; \alpha \; t}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}}} \right)^{{\alpha}\; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}}} \right)^{\; {\pi/2}}^{{- }\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}}} \right)^{{- }\; \alpha \; t}} - {\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}}} \right)^{{\pi}/2}{^{{- }\; \alpha \; t}.}}}}}}}}} & (14)\end{matrix}$

After FT, one obtains for (c₊₁,c⁻¹)-sampling:

$\begin{matrix}{{{\lambda_{{+ 1},{- 1}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}} \right)}};}{\lambda_{{+ 1},{- 1}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}}} \right)}}{{\lambda_{{+ 1},{- 1}}^{A -} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}}} \right)}};}{\lambda_{{+ 1},{- 1}}^{D -} = {{- \frac{1}{2}}\left( {{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}}} \right)}}{{{or}\mspace{14mu} {equivalently}},{\Phi_{+ 1} = {{{arc}\; {\cos \left( {\lambda_{{+ 1},{- 1}}^{A +} - \lambda_{{+ 1},{- 1}}^{D -}} \right)}} = {{arc}\; {\sin \left( {\lambda_{{+ 1},{- 1}}^{D +} - \lambda_{{+ 1},{- 1}}^{A -}} \right)}}}}}\Phi_{- 1} = {{{arc}\; {\cos \left( {\lambda_{{+ 1},{- 1}}^{A +} + \lambda_{{+ 1},{- 1}}^{D -}} \right)}} = {{- {arc}}\; {{\sin \left( {\lambda_{{+ 1},{- 1}}^{D +} + \lambda_{{+ 1},{- 1}}^{A -}} \right)}.}}}} & (15)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1}}^{D} = \frac{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}}}{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}}} + {{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}}}}}}{M_{{+ 1},{- 1}}^{Q} = \frac{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}}} + {{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}}} + {{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}}}} +} \\{{{{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}}}} + {{{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}}}}}\end{matrix}}}M_{{+ 1},{- 1}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}}}.}}} & (16)\end{matrix}$

(c₊₃,c⁻³)-Sampling (PMS)

The two interferograms for (c₊₃,c⁻³)-PMS are given by

$\begin{matrix}\begin{matrix}{{C_{{+ 3},{- 3}}(t)} = \begin{bmatrix}{c_{+ 3}(t)} \\{c_{- 3}(t)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \frac{3\pi}{4} + \Phi_{+ 3}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{3\pi}{4} + \Phi_{- 3}} \right)}\end{bmatrix}} \\{= {\begin{bmatrix}{- {\sin \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 3}} \right)}} \\{- {\sin \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 3}} \right)}}\end{bmatrix}.}}\end{matrix} & (17)\end{matrix}$

so that the resulting signal S_(+3,−3)(t) is proportional to

$\begin{matrix}{{{S_{{+ 3},{- 3}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 3},{- 3}}{C_{{+ 3},{- 3}}(t)}}} = {{{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 3}(t)} \\{c_{- 3}(t)}\end{bmatrix}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}{{- 1} - } & {{- 1} + }\end{bmatrix}}\begin{bmatrix}{- {\sin \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 3}} \right)}} \\{- {\sin \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 3}} \right)}}\end{bmatrix}} = {{{\frac{1}{\sqrt{2}}\left( {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) + {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right){\cos \left( {\alpha \; t} \right)}} + {\frac{1}{\sqrt{2}}\left( {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) + {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\frac{1}{\sqrt{2}}\left( {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) + {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right)\frac{^{{\alpha}\; t} + ^{{- }\; \alpha \; t}}{2}} + {\frac{1}{\sqrt{2}}\left( {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) + {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right)\frac{^{\; \alpha \; t} - ^{{- }\; \alpha \; t}}{2}}} = {{{\frac{1}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}} + {\sin \; \Phi_{- 3}} + {\cos \; \Phi_{+ 3}} - {\sin \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}} - {\sin \; \Phi_{- 3}}} \right)^{\; \alpha \; t}} + {\frac{1}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}} + {\sin \; \Phi_{- 3}} - {\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}} + {\sin \; \Phi_{- 3}}} \right)^{{- }\; \alpha \; t}} + {\frac{}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}} - {\sin \; \Phi_{- 3}} - {\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}} - {\sin \; \Phi_{- 3}}} \right)^{\; \alpha \; t}} + {\frac{}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}} - {\sin \; \Phi_{- 3}} + {\cos \; \Phi_{+ 3}} - {\sin \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}} + {\sin \; \Phi_{- 3}}} \right)^{{- }\; \alpha \; t}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)^{\; {\pi/2}}^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 3}} + {\sin \; \Phi_{- 3}}} \right)^{{- {\alpha}}\; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}}} \right)^{\; {\pi/2}}{^{\; \alpha \; t}.}}}}}}}}} & (18)\end{matrix}$

After FT, one obtains for (c₊₃,c⁻³)-sampling:

$\begin{matrix}{{{\lambda_{{+ 3},{- 3}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right)}};}{\lambda_{{+ 3},{- 3}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)}}{{\lambda_{{+ 3},{- 3}}^{A -} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 3}} + {\sin \; \Phi_{- 3}}} \right)}};}{\lambda_{{+ 3},{- 3}}^{D -} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}}} \right)}}{{{or}\mspace{14mu} {equivalently}},{\Phi_{+ 3} = {{{arc}\; {\cos \left( {\lambda_{{+ 3},{- 3}}^{A +} + \lambda_{{+ 3},{- 3}}^{D -}} \right)}} = {{arc}\; {\sin \left( {\lambda_{{+ 3},{- 3}}^{D +} + \lambda_{{+ 3},{- 3}}^{A -}} \right)}}}}}{\Phi_{- 3} = {{{arc}\; {\cos \left( {\lambda_{{+ 3},{- 3}}^{A +} - \lambda_{{+ 3},{- 3}}^{D -}} \right)}} = {{- {arc}}\; {{\sin \left( {\lambda_{{+ 3},{- 3}}^{D +} - \lambda_{{+ 3},{- 3}}^{A -}} \right)}.}}}}} & (19)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 3},{- 3}}^{D} = \frac{{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}}{{{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}} + {{{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}}}}{M_{{+ 3},{- 3}}^{Q} = \frac{{{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}} + {{{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}} + {{{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}} +} \\{{{{\sin \; \Phi_{+ 3}} + {\sin \; \Phi_{- 3}}}} + {{{\cos \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}}}}}\end{matrix}}}{M_{{+ 3},{- 3}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}}.}}}} & (20)\end{matrix}$

(c₊₁,c⁻¹,c₊₃,c⁻³)-Sampling (DPMS)

Addition of S_(+1,−1)(t) and S_(+3,−3)(t) yields

$\begin{matrix}{{S_{{+ 1},{- 1},{+ 3},{- 3}}(t)} = {{{S_{{+ 1},{- 1}}(t)} + {S_{{+ 3},{- 3}}(t)}} \propto {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)^{\; {\pi/2}}^{\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}} - {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)^{{- }\; \alpha \; t}} - {\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} - {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right)^{\; {\pi/2}}{^{{- }\; \alpha \; t}.}}}}} & (21)\end{matrix}$

After FT, one obtains for (c₊₁,c⁻¹,c₊₃,c⁻³)-sampling:

$\begin{matrix}{{{\lambda_{{+ 1},{- 1},{+ 3},{- 3}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right)}};}{\lambda_{{+ 1},{- 1},{+ 3},{- 3}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)}}{{\lambda_{{+ 1},{- 1},{+ 3},{- 3}}^{A -} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}} - {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)}};}{\lambda_{{+ 1},{- 1},{+ 3},{- 3}}^{D -} = {{- \frac{1}{2}}{\left( {{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} - {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right).}}}} & (22)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1},{+ 3},{- 3}}^{D} = \frac{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}} +} \\{{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}}\end{matrix}}}{M_{{+ 1},{- 1},{+ 3},{- 3}}^{Q} = \frac{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}} + {{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}} +} \\{{{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}} + {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}} +} \\{{{{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}} - {\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}} +} \\{{{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} - {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}}\end{matrix}}}{M_{{+ 1},{- 1},{+ 3},{- 3}}^{A} = {\frac{1}{4}{{{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} + {\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}}.}}}} & (23)\end{matrix}$

0 and π/2-Shifted Mirrored Sampling

(c₊₀,c⁻²)-Sampling (PMS)

The two interferograms for (c₊₀,c⁻²)-PMS are given by

$\begin{matrix}\begin{matrix}{{C_{{+ 0},{- 2}}(t)} = \begin{bmatrix}{c_{+ 0}(t)} \\{c_{- 2}(t)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \Phi_{+ 0}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{2} + \Phi_{- 2}} \right)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \Phi_{+ 0}} \right)} \\{- {\sin \left( {{{- \alpha}\; t} + \Phi_{- 2}} \right)}}\end{bmatrix}} \\{= {\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \Phi_{+ 0}} \right)} \\{\sin \left( {{{+ \alpha}\; t} - \Phi_{- 2}} \right)}\end{bmatrix}.}}\end{matrix} & (24)\end{matrix}$

so that the resulting signal S_(+0,−2)(t) is proportional to

$\begin{matrix}{{{S_{{+ 0},{- 2}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 0},{- 2}}{C_{{+ 0},{- 2}}(t)}}} = {{{\begin{bmatrix}1 & \end{bmatrix}\left\lbrack \begin{matrix}1 & 0 \\0 & 1\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}{c_{+ 0}(t)} \\{c_{- 2}(t)}\end{matrix} \right\rbrack} = {\quad{{\left\lbrack \begin{matrix}1 & \end{matrix} \right\rbrack \begin{bmatrix}c_{+ 0} \\c_{- 2}\end{bmatrix}} = {{\begin{bmatrix}1 & \end{bmatrix}\left\lbrack \begin{matrix}{\cos \left( {{\alpha \; t} + \Phi_{+ 0}} \right)} \\{\sin \left( {{\alpha \; t} - \Phi_{- 2}} \right)}\end{matrix} \right\rbrack} = {{\left( {{\cos \; \Phi_{+ 0}{\cos \left( {\alpha \; t} \right)}} - {\sin \; \Phi_{+ 0}{\sin \left( {\alpha \; t} \right)}}} \right) - {\left( {{\sin \; \Phi_{- 2}{\cos \left( {\alpha \; t} \right)}} - {\cos \; \Phi_{- 2}{\sin \left( {\alpha \; t} \right)}}} \right)}} = {{{\left( {{\cos \; \Phi_{+ 0}} - {\; \sin \; \Phi_{- 2}}} \right){\cos \left( {\alpha \; t} \right)}} - {\left( {{\sin \; \Phi_{+ 0}} - {\; \cos \; \Phi_{- 2}}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\left( {{\cos \; \Phi_{+ 0}} - {\; \sin \; \Phi_{- 2}}} \right)\frac{^{\; \alpha \; t} + ^{{- }\; \alpha \; t}}{2}} - {\left( {{\sin \; \Phi_{+ 0}} - {\; \cos \; \Phi_{- 2}}} \right) \frac{^{\; \alpha \; t} - ^{{- }\; \alpha \; t}}{2}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} - {\quad {{\frac{}{2} \left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} - {\quad {\frac{1}{2} \left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}{^{{- }\; \alpha \; t}.}}}}}}}}}}}}}}} & (25)\end{matrix}$

After FT, one obtains for (c₊₀,c⁻²)-sampling:

$\begin{matrix}{{{\lambda_{{+ 0},{- 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}} \right)}};}{\lambda_{{+ 0},{- 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}} \right)}}{{\lambda_{{+ 0},{- 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}}} \right)}};}{\lambda_{{+ 0},{- 2}}^{D -} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}}} \right)}}{{{or}\mspace{14mu} {equivalently}},{\Phi_{+ 0} = {{{arc}\; {\cos \left( {\lambda_{{+ 0},{- 2}}^{A +} + \lambda_{{+ 0},{- 2}}^{A -}} \right)}} = {{arc}\; {\sin \left( {\lambda_{{+ 0},{- 2}}^{D +} - \lambda_{{+ 0},{- 2}}^{D -}} \right)}}}}}{{\Phi_{- 2} = {{{arc}\; {\cos \left( {\lambda_{{+ 0},{- 2}}^{A +} - \lambda_{{+ 0},{- 2}}^{A -}} \right)}} = {{- {arc}}\; {\sin \left( {\lambda_{{+ 0},{- 2}}^{D +} + \lambda_{{+ 0},{- 2}}^{D -}} \right)}}}},}} & (26)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}}}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}}} + {{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}}}}}{M_{{+ 0},{- 2}}^{Q} = \frac{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}}} + {{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}}} + {{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}}} +} \\{{{{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}}}} + {{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}}}}}\end{matrix}}}{M_{{+ 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}}}.}}}} & (27)\end{matrix}$

(c⁻⁰,c₊₂)-Sampling (PMS)

The two interferograms for (c⁻⁰,c₊₂)-PMS are given by

$\begin{matrix}\begin{matrix}{{C_{{- 0},{+ 2}}(t)} = \begin{bmatrix}c_{- 0} \\c_{+ 2}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{- \alpha}\; t} + \Phi_{- 0}} \right)} \\{\cos \left( {{{+ \alpha}\; t} + \frac{\pi}{2} + \Phi_{+ 2}} \right)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \left( {{{- \alpha}\; t} + \Phi_{- 0}} \right)} \\{- {\sin \left( {{{+ \alpha}\; t} + \Phi_{+ 2}} \right)}}\end{bmatrix}} \\{= {\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} - \Phi_{- 0}} \right)} \\{\sin \left( {{{+ \alpha}\; t} + \Phi_{+ 2}} \right)}\end{bmatrix}.}}\end{matrix} & (28)\end{matrix}$

so that the resulting signal S_(−0,+2)(t) is proportional to

$\begin{matrix}{{{S_{{- 0},{+ 2}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{- 0},{+ 2}}{C_{{- 0},{+ 2}}(t)}}} = {{{\begin{bmatrix}1 & \end{bmatrix}\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}{c_{- 0}(t)} \\{c_{+ 2}(t)}\end{matrix} \right\rbrack} = {{\quad{{\left\lbrack \begin{matrix}1 & {- }\end{matrix} \right\rbrack \begin{bmatrix}c_{- 0} \\c_{+ 2}\end{bmatrix}} = {{\begin{bmatrix}1 & {- }\end{bmatrix}\left\lbrack \begin{matrix}{\cos \left( {{\alpha \; t} - \Phi_{- 0}} \right)} \\{- {\sin \left( {{\alpha \; t} + \Phi_{+ 2}} \right)}}\end{matrix} \right\rbrack} = {{\left( {{\cos \; \Phi_{- 0}{\cos \left( {\alpha \; t} \right)}} + {\sin \; \Phi_{- 0}{\sin \left( {\alpha \; t} \right)}}} \right) + {\left( {{\sin \; \Phi_{+ 2}{\cos \left( {\alpha \; t} \right)}} + {\cos \; \Phi_{- 2}{\sin \left( {\alpha \; t} \right)}}} \right)}} = {{{\left( {{\cos \; \Phi_{- 0}} + {\; \sin \; \Phi_{+ 2}}} \right){\cos \left( {\alpha \; t} \right)}} + {\left( {{\sin \; \Phi_{- 0}} + {\; \cos \; \Phi_{+ 2}}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\left( {{\cos \; \Phi_{- 0}} + {\; \sin \; \Phi_{+ 2}}} \right)\frac{^{\; \alpha \; t} + ^{{- }\; \alpha \; t}}{2}} + {\left( {{\sin \; \Phi_{- 0}} + {\; \cos \; \Phi_{+ 2}}} \right)\frac{^{\; \alpha \; t} - ^{{- }\; \alpha \; t}}{2}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}} \right)^{\; \alpha \; t}} - {\frac{}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}} \right) ^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}} \right)^{{- }\; \alpha \; t}} +}}}}}}\quad}{\quad{{\frac{}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}} \right)^{{- }\; \alpha \; t}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}} \right)^{\; \alpha \; t}} - {\quad {{\frac{1}{2} \left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}} \right)^{\frac{\pi}{2}}^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}} \right)^{{- }\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}} \right)^{\frac{\pi}{2}}{^{{- }\; \alpha \; t}.}}}}}}}}}} & (29)\end{matrix}$

After FT, one obtains for (c⁻⁰,c₊₂)-sampling:

$\begin{matrix}{{{{\lambda_{{- 0},{+ 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}} \right)}};}{\lambda_{{- 0},{+ 2}}^{D +} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}} \right)}}{{\lambda_{{- 0},{+ 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}} \right)}};}\lambda_{{- 0},{+ 2}}^{D -} = {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}} \right)}}{{{or}\mspace{14mu} {equivalently}},\begin{matrix}{\Phi_{- 0} = {\arccos \left( {\lambda_{{- 0},{+ 2}}^{A +} + \lambda_{{- 0},{+ 2}}^{A -}} \right)}} \\{= {- {\arcsin \left( {\lambda_{{- 0},{+ 2}}^{D +} - \lambda_{{- 0},{+ 2}}^{D -}} \right)}}} \\{\Phi_{+ 2} = {\arccos \left( {\lambda_{{- 0},{+ 2}}^{A +} - \lambda_{{- 0},{+ 2}}^{A -}} \right)}} \\{= {{\arcsin \left( {\lambda_{{- 0},{+ 2}}^{D +} + \lambda_{{- 0},{+ 2}}^{D -}} \right)}.}}\end{matrix}}} & (30)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}}{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}} + {{{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}}}}}{M_{{- 0},{+ 2}}^{Q} = \frac{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}} + {{{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}} + {{{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}}} +} \\{{{{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}}} + {{{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}}}}\end{matrix}}}{M_{{- 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}}.}}}} & (31)\end{matrix}$

(c₊₀,c⁻²,c⁻⁰,c₊₂)-Sampling (DPMS)

Addition of S_(+0,−2)(t) and S_(−0,+2)(t) yields

$\begin{matrix}{{S_{{+ 0},{- 2},{- 0},{+ 2}}(t)} = {{{S_{{+ 0},{- 2}}(t)} + {S_{{- 0},{+ 2}}(t)}} \propto {{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} +} \\{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}\end{pmatrix}^{{\alpha}\; t}} + {\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}} -} \\{{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}}\end{pmatrix}^{\frac{\pi}{2}}^{{\alpha}\; t}} + {\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}} +} \\{{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}}\end{pmatrix}^{{- {\alpha}}\; t}} - {\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}} -} \\{{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}}\end{pmatrix}^{\frac{\pi}{2}}{^{{- {\alpha}}\; t}.}}}}} & (32)\end{matrix}$

After FT, one obtains for (c₊₀,c⁻²,c⁻⁰,c₊₂)-sampling:

$\begin{matrix}{{{\lambda_{{+ 0},{- 2},{- 0},{+ 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}} \right)}};}{\lambda_{{+ 0},{- 2},{- 0},{+ 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}} \right)}}{{\lambda_{{+ 0},{- 2},{- 0},{+ 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}} \right)}};}{\lambda_{{+ 0},{- 2},{- 0},{+ 2}}^{D -} = {{- \frac{1}{2}}{\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}} \right).}}}} & (33)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2},{- 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}}{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}} +} \\{{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}}}\end{matrix}}}{M_{{+ 0},{- 2},{- 0},{+ 2}}^{Q} = \frac{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}} +} \\{{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}}}\end{matrix}}{\begin{matrix}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}} +} \\{{{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}}} +} \\{{{{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}}} +} \\{{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}}}\end{matrix}}}{M_{{+ 0},{- 2},{- 0},{+ 2}}^{A} = {\frac{1}{4}{{{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}}.}}}} & (34)\end{matrix}$

Application for Partially Identical Secondary Phase Shifts

Yet another embodiment of the present invention relates to cleanabsorption mode NMR data acquisition for partially identical secondaryphase shifts. In the embodiment described below, forward and backwardsampling for given n are associated with the same secondary phase shift.If forward and backward sampling of the time domain for given n areassociated with the same secondary phase shift, one can define

Φ_(n)=Φ_(+n)=Φ_(−n)  (35)

In the following, the coefficient vector elements and the figures ofmerit for the different sampling schemes described above are simplifiedusing Equation 35.

‘States’ Sampling

(c₊₀,c₊₂)-Sampling

Given Eq. 4, one obtains for (c₊₀,c₊₂)-sampling, under the condition ofEq. 35:

$\begin{matrix}{{{\lambda_{{+ 0},{+ 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{+ 0},{+ 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}} \right)}}{{\lambda_{{+ 0},{+ 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{+ 0},{+ 2}}^{D -} = {{- \frac{1}{2}}{\left( {{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}} \right).}}}} & (36)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}}}}{M_{{+ 0},{+ 2}}^{Q} = \frac{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}} +} \\{{{{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}}}\end{matrix}}}{M_{{+ 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}.}}}} & (37)\end{matrix}$

One embodiment of this aspect of the present invention involves backwardsampling which is described as follows:

(c⁻⁰,c⁻²)-Sampling

Given Eq. 8, one obtains for (c⁻⁰,c⁻²)-sampling, under the condition ofEq. 35:

$\begin{matrix}{{{\lambda_{{- 0},{- 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{- 0},{- 2}}^{D +} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}} \right)}}{{\lambda_{{- 0},{- 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{- 0},{- 2}}^{D -} = {\frac{1}{2}{\left( {{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}} \right).}}}} & (38)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}}}}{M_{{- 0},{- 2}}^{Q} = \frac{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}} +} \\{{{{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}}}\end{matrix}}}{M_{{- 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}.}}}} & (39)\end{matrix}$

Dual ‘States’ Sampling

(c₊₀,c₊₂,c⁻⁰,c⁻²)-Sampling

Given Eq. 11, one obtains for (c₊₀,c₊₂,c⁻⁰,c⁻²)-sampling, under thecondition of Eq. 35:

$\begin{matrix}{{{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{A +} = \left( {{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}} \right)};}{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{D +} = 0}{{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{A -} = \left( {{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}} \right)};}{\lambda_{{+ 0},{+ 2},{- 0},{- 2}}^{D -} = 0.}} & (40)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{+ 2},{- 0},{- 2}}^{D} = 1}{M_{{+ 0},{+ 2},{- 0},{- 2}}^{Q} = \frac{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}}}}}{M_{{+ 0},{+ 2},{- 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}.}}}} & (41)\end{matrix}$

π/4 and 3π/4-Shifted Mirrored Sampling

(c₊₁,c⁻¹)-Sampling (PMS)

Given Eq. 15, one obtains for (c₊₁,c⁻¹)-sampling, under the condition ofEq. 35:

λ_(−1,−1) ^(A+)=cos Φ₁; λ_(+1,−1) ^(D+)=0

λ_(+1,−1) ^(A−)=−sin Φ₁; λ_(+1,−1) ^(D−)=0  (42).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1}}^{D} = 1}{M_{{+ 1},{- 1}}^{Q} = \frac{{\cos \; \Phi_{1}}}{{{\cos \; \Phi_{1}}} + {{\sin \; \Phi_{1}}}}}{M_{{+ 1},{- 1}}^{A} = {{{\cos \; \Phi_{1}}}.}}} & (43)\end{matrix}$

(c₊₃,c⁻³)-Sampling (PMS)

Given Eq. 19, one obtains for (c₊₃,c⁻³)-sampling, under the condition ofEq. 35:

λ_(+3,−3) ^(A+)=cos Φ₃; λ_(+3,−3) ^(D+)=0

λ_(+3,−3) ^(A−)=−sin Φ₃; λ_(+3,−3) ^(D−)=0  (44).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 3},{- 3}}^{D} = 1}{M_{{+ 3},{- 3}}^{Q} = \frac{{\cos \; \Phi_{3}}}{{{\cos \; \Phi_{3}}} + {{\sin \; \Phi_{3}}}}}{M_{{+ 3},{- 3}}^{A} = {{{\cos \; \Phi_{3}}}.}}} & (45)\end{matrix}$

(c₊₁,c⁻¹,c₊₃,c⁻³)-Sampling (DPMS)

Given Eq. 22, one obtains for (C₊₁,c⁻¹,c₊₃,c⁻³)-sampling, under thecondition of Eq. 35:

λ_(+1,−1, +3, −3) ^(A+)=cos Φ₁+cos Φ₃; λ_(+1,−1, +3, −3) ^(D+)=0

λ_(+1,−1) ^(A−)=−(sin Φ₁−Φ₃); λ_(+1,−1, +3, −3) ^(D−)=0  (46).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1},{+ 3},{- 3}}^{D} = 1}{M_{{+ 1},{- 1},{+ 3},{- 3}}^{Q} = \frac{{{\cos \; \Phi_{1}} + {\cos \; \Phi_{3}}}}{{{{\cos \; \Phi_{1}}\; + {\cos \; \Phi_{3}}}} + {{{\sin \; \Phi_{1}} - {\sin \; \Phi_{3}}}}}}{M_{{+ 1},{- 1},{+ 3},{- 3}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{1}} + {\cos \; \Phi_{3}}}}.}}}} & (47)\end{matrix}$

0 and π/2-Shifted Mirrored Sampling

(c₊₀,c⁻²)-Sampling (PMS)

Given Eq. 26, one obtains for (c₊₀,c⁻²)-sampling, under the condition ofEq. 35:

$\begin{matrix}{{{\lambda_{{+ 0},{- 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{+ 0},{- 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}} \right)}}{{\lambda_{{+ 0},{- 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{+ 0},{- 2}}^{D -} = {{- \frac{1}{2}}{\left( {{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}} \right).}}}} & (48)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}}}}{M_{{+ 0},{- 2}}^{Q} = \frac{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}} +} \\{{{{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}}}\end{matrix}}}{M_{{+ 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}.}}}} & (49)\end{matrix}$

(c⁻⁰,c₊₂)-Sampling (PMS)

Given Eq. 30, one obtains for (c⁻⁰,c₊₂)-sampling, under the condition ofEq. 35:

$\begin{matrix}{{{\lambda_{{- 0},{+ 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{- 0},{+ 2}}^{D +} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}} \right)}}{{\lambda_{{- 0},{+ 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}} \right)}};}{\lambda_{{- 0},{+ 2}}^{D -} = {\frac{1}{2}{\left( {{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}} \right).}}}} & (50)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}}}}{M_{{- 0},{+ 2}}^{Q} = \frac{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}}}{\begin{matrix}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} - {\sin \; \Phi_{2}}}} +} \\{{{{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}}} + {{{\sin \; \Phi_{0}} + {\sin \; \Phi_{2}}}}}\end{matrix}}}{M_{{- 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}.}}}} & (51)\end{matrix}$

(c₊₀,c⁻²,c⁻⁰,c₊₂)-Sampling (DPMS)

Given Eq. 33, one obtains for (c₊₀,c⁻², c⁻⁰,c₊₂)-sampling, under thecondition of Eq. 35:

λ_(+0,−2, −0, −2) ^(A+)=cos Φ₀+cos Φ₂; λ_(+0,−2, +0, −2) ^(D+)=0

λ_(+0,−2, −0, −2) ^(A−)=cos Φ₀−cos Φ₂; λ_(+0,−2, −0, −2) ^(D−)=0  (52).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2},{- 0},{+ 2}}^{D} = 1}{M_{{+ 0},{- 2},{- 0},{+ 2}}^{Q} = \frac{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}} + {{{\cos \; \Phi_{0}} - {\cos \; \Phi_{2}}}}}}{M_{{+ 0},{- 2},{- 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{0}} + {\cos \; \Phi_{2}}}}.}}}} & (53)\end{matrix}$

Another aspect of the present invention relates to a general applicationfor partly identical secondary phase shifts.

Given that two forward and two backward sampling are associated withsecondary phase shifts independent of n, one can define

Φ₊=Φ_(+n)

Φ₌=Φ_(−n)  (54).

In the following, the coefficient vector elements and the figures ofmerit for the different sampling schemes of the above sections aresimplified using Eq. 54.

‘States’ Sampling

(c₊₀,c₊₂)-Sampling

Given Eq. 4, one obtains for (c₊₀,c₊₂)-sampling, under the condition ofEq. 54:

λ_(+2,+2) ^(A+)=cos Φ₊; λ_(+0,−2) ^(D+)=sin Φ₊

λ_(+0,−2) ^(A−)=0; λ_(+0,+2) ^(D−)=0  (55),

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{+ 2}}^{D} = \frac{{\cos \; \Phi_{+}}}{{{\cos \; \Phi_{+}}} + {{\sin \; \Phi_{+}}}}}{M_{{+ 0},{+ 2}}^{Q} = 1}{M_{{+ 0},{+ 2}}^{A} = {{{\cos \; \Phi_{+}}}.}}} & (56)\end{matrix}$

(c⁻⁰,c⁻²)-Sampling

Given Eq. 8, one obtains for (c⁻⁰,c⁻²)-sampling, under the condition ofEq. 54:

λ_(+3,−3) ^(A+)=cos Φ₃; λ_(+3,−3) ^(D+)=0

λ_(+3,−3) ^(A−)=−sin Φ₃; λ_(+3,−3) ^(D−)=0  (57).

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{- 2}}^{D} = \frac{{\cos \; \Phi_{-}}}{{{\cos \; \Phi_{-}}} + {{\sin \; \Phi_{-}}}}}{M_{{- 0},{- 2}}^{Q} = 1}{M_{{- 0},{- 2}}^{A} = {{{\cos \; \Phi_{-}}}.}}} & (58)\end{matrix}$

(c₊₀,c₊₂,c⁻⁰,c⁻²)-Sampling

Given Eq. 11, one obtains for (c₊₀,c₊₂,c⁻⁰,c⁻²)-sampling, under thecondition of Eq. 54:

λ_(+0,+2,−0,−2) ^(A+)=cos Φ₊+cos Φ⁻; λ_(+0,+2,−0,−2) ^(D+)=sin Φ₊−sin Φ⁻

λ_(+0,+2,−0,−2) ^(A−)=0; λ_(+0,+2,−0,−2) ^(D−)=0  (59).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{+ 2},{- 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}}{M_{{+ 0},{+ 2},{- 0},{- 2}}^{Q} = 1}{M_{{+ 0},{+ 2},{- 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}.}}}} & (60)\end{matrix}$

π/4 and 3π/4-Shifted Mirrored Sampling

(c₊₁,c⁻¹)-Sampling (PMS)

Given Eq. 15, one obtains for (c₊₁,c⁻¹)-sampling, under the condition ofEq. 54:

$\begin{matrix}{{{\lambda_{{+ 1},{- 1}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}} \right)}};}{\lambda_{{+ 1},{- 1}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}} \right)}}{{\lambda_{{+ 1},{- 1}}^{A -} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{+}} + {\sin \; \Phi_{-}}} \right)}};}{\lambda_{{+ 1},{- 1}}^{D -} = {{- \frac{1}{2}}{\left( {{\cos \; \Phi_{+}} - {\cos \; \Phi_{-}}} \right).}}}} & (61)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1}}^{D} = \frac{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}}{M_{{+ 1},{- 1}}^{Q} = \frac{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}} +} \\{{{{\sin \; \Phi_{+}} + {\sin \; \Phi_{-}}}} + {{{\cos \; \Phi_{+}} - {\cos \; \Phi_{-}}}}}\end{matrix}}}{M_{{+ 1},{- 1}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}.}}}} & (62)\end{matrix}$

(c₊₃,c⁻³)-Sampling (PMS)

Given Eq. 19, one obtains for (c₊₃,c⁻³)-sampling, under the condition ofEq. 54:

$\begin{matrix}{{{\lambda_{{+ 3},{- 3}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}} \right)}};}{\lambda_{{+ 3},{- 3}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}} \right)}}{{\lambda_{{+ 3},{- 3}}^{A -} = {\frac{1}{2}\left( {{\sin \; \Phi_{+}} + {\sin \; \Phi_{-}}} \right)}};}{\lambda_{{+ 3},{- 3}}^{D -} = {\frac{1}{2}{\left( {{\cos \; \Phi_{+}} - {\cos \; \Phi_{-}}} \right).}}}} & (63)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 3},{- 3}}^{D} = \frac{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}{{{{\cos \; \Phi_{-}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}}{M_{{+ 3},{- 3}}^{Q} = \frac{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}} +} \\{{{{\sin \; \Phi_{+}} + {\sin \; \Phi_{-}}}} + {{{\cos \; \Phi_{+}} - {\cos \; \Phi_{-}}}}}\end{matrix}}}{{M_{{+ 3},{- 3}}^{A} = {\frac{1}{2}{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}}},}} & (64)\end{matrix}$

(c₊₁,c⁻¹,c₊₃,c⁻³)-Sampling (DPMS)

Given Eq. 22, one obtains for (C+1,c⁻¹,C₊₃,c⁻³)-sampling, under thecondition of Eq. 54:

λ_(+1,−1,+3,−3) ^(A+)=cos Φ₊+cos Φ⁻; λ_(+1,−1,+3,−3) ^(D+)=sin Φ₊−sin Φ⁻

λ_(+1,−1,+3,−3) ^(A−)=0; λ_(+1,−1,+3,−3) ^(D−)=0  (65).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1},{+ 3},{- 3}}^{D} = \frac{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}}{M_{{+ 1},{- 1},{+ 3},{- 3}}^{Q} = 1}{M_{{+ 1},{- 1},{+ 3},{- 3}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}.}}}} & (66)\end{matrix}$

0 and π/2-Shifted Mirrored Sampling

(c₊₀,c⁻²)-Sampling (PMS)

Given Eq. 26, one obtains for (C₊₀,c⁻²)-sampling, under the condition ofEq. 54:

$\begin{matrix}{{{\lambda_{{+ 0},{- 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}} \right)}};}{\lambda_{{+ 0},{- 2}}^{D +} = {\frac{1}{2}\left( {{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}} \right)}}{{\lambda_{{+ 0},{- 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{+}} - {\cos \; \Phi_{-}}} \right)}};}{\lambda_{{+ 0},{- 2}}^{D -} = {{- \frac{1}{2}}{\left( {{\sin \; \Phi_{+}} + {\sin \; \Phi_{-}}} \right).}}}} & (67)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2}}^{D} = \frac{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}}{M_{{+ 0},{- 2}}^{Q} = \frac{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}{\begin{matrix}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}} +} \\{{{{\cos \; \Phi_{+}} - {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} + {\sin \; \Phi_{-}}}}}\end{matrix}}}M_{{+ 0},{- 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}.}}} & (68)\end{matrix}$

(c⁻⁰,c₊₂)-Sampling (PMS)

Given Eq. 30, one obtains for (c⁻⁰,c₊₂)-sampling, under the condition ofEq. 54:

$\begin{matrix}{{{\lambda_{{- 0},{+ 2}}^{A +} = {\frac{1}{2}\left( {{\cos \; \Phi_{-}} + {\cos \; \Phi_{+}}} \right)}};}{\lambda_{{- 0},{+ 2}}^{D +} = {{- \frac{1}{2}}\left( {{\sin \; \Phi_{-}} - {\sin \; \Phi_{+}}} \right)}}{{\lambda_{{- 0},{+ 2}}^{A -} = {\frac{1}{2}\left( {{\cos \; \Phi_{-}} - {\cos \; \Phi_{+}}} \right)}};}{\lambda_{{- 0},{+ 2}}^{D -} = {\frac{1}{2}{\left( {{\sin \; \Phi_{-}} + {\sin \; \Phi_{+}}} \right).}}}} & (69)\end{matrix}$

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{-}} + {\cos \; \Phi_{+}}}}{{{{\cos \; \Phi_{-}} + {\cos \; \Phi_{+}}}} + {{{\sin \; \Phi_{-}} - {\sin \; \Phi_{+}}}}}}{M_{{- 0},{+ 2}}^{Q} = \frac{{{{\cos \; \Phi_{-}} + {\cos \; \Phi_{+}}}} + {{{\sin \; \Phi_{-}} - {\sin \; \Phi_{+}}}}}{\begin{matrix}{{{{\cos \; \Phi_{-}} + {\cos \; \Phi_{+}}}} + {{{\sin \; \Phi_{-}} - {\sin \; \Phi_{+}}}} +} \\{{{{\cos \; \Phi_{-}} - {\cos \; \Phi_{+}}}} + {{{\sin \; \Phi_{-}} + {\sin \; \Phi_{+}}}}}\end{matrix}}}M_{{- 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{-}} + {\cos \; \Phi_{+}}}}.}}} & (70)\end{matrix}$

(c₊₀,c⁻²,c⁻⁰,c₊₂)-Sampling (DPMS)

Given Eq. 33, one obtains for (c₊₀,c⁻², c⁻⁰,c₊₂)-sampling, under thecondition of Eq. 54:

λ_(+0,−2,−0,+2) ^(A+)=cos Φ₊+cos Φ⁻; λ_(+0,−2,−0,+2) ^(D+)=sin Φ₊−sin Φ⁻

λ_(+0,−2,−0,+2) ^(A−)=0; λ_(+0,−2,−0,+2) ^(D−)=0  (71).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2},{- 0},{+ 2}}^{D} = \frac{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}} + {{{\sin \; \Phi_{+}} - {\sin \; \Phi_{-}}}}}}{M_{{+ 0},{- 2},{- 0},{+ 2}}^{Q} = 1}{M_{{+ 0},{- 2},{- 0},{+ 2}}^{A} = {\frac{1}{2}{{{{\cos \; \Phi_{+}} + {\cos \; \Phi_{-}}}}.}}}} & (72)\end{matrix}$

Application for Identical Secondary Phase Shifts

One embodiment of the present invention relates to clean absorption modeNMR data acquisition for identical secondary phase shifts.

Given that secondary phase shifts of forward and backward sampling arethe same and independent of n, one can define

Φ=Φ_(±n)  (73).

In the following, the coefficient vector elements and the figures ofmerit for the different sampling schemes of the above sections aresimplified using Eq. 73.

‘States’ Sampling

(c₊₀,c₊₂)-Sampling

Given Eq. 4, one obtains for (c₊₀,c₊₂)-sampling, under the condition ofEq. 73:

λ_(+0,+2) ^(A+)=cos Φ; λ_(+0,+2) ^(D+)=sin Φ

λ_(+0,+2) ^(A−)=0; λ_(+0,+2) ^(D−)=0  (74).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{+ 2}}^{D} = \frac{{\cos \; \Phi}\; }{{{\cos \; \Phi}} + {{\sin \; \Phi}}}}{M_{{+ 0},{+ 2}}^{Q} = 1}{M_{{+ 0},{+ 2}}^{A} = {{{\cos \; \Phi}}.}}} & (75)\end{matrix}$

(c⁻⁰,c⁻²)-Sampling

Given Eq. 8, one obtains for (c⁻⁰,c⁻²)-sampling, under the condition ofEq. 73:

λ_(−0,−2) ^(A+)=cos Φ; λ_(−0,−2) ^(D+)=sin Φ

λ_(−0,−2) ^(A−)=0; λ_(−0,−2) ^(D−)=0  (76).

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{- 2}}^{D} = \frac{{\cos \; \Phi}\; }{{{\cos \; \Phi}} + {{\sin \; \Phi}}}}{M_{{- 0},{- 2}}^{Q} = 1}{M_{{- 0},{- 2}}^{A} = {{{\cos \; \Phi}}.}}} & (77)\end{matrix}$

(c₊₀,c₊₂,c⁻⁰,c⁻²)-Sampling

Given Eq. 11, one obtains for (c₊₀,c₊₂,c⁻⁰,c⁻²)-sampling, under thecondition of Eq. 73:

λ_(+0,+2,−0,−2) ^(A+)=cos Φ; λ_(+0,+2,−0,−2) ^(D+)=0

λ_(+0,+2,−0,−2) ^(A−)=0; λ_(+0,+2,−0,−2) ^(D−)=0  (78).

so that the figures of merit are given by

M _(+0,+2,−0,−2) ^(D)=1

M _(+0,+2,−0,−2) ^(Q)=1

M _(+0,+2,−0,−2) ^(A)=|cos Φ|  (79).

π/4 and 3π/4-Shifted Mirrored Sampling

(c₊₁,c⁻¹)-Sampling (PMS)

Given Eq. 15, one obtains for (c₊₁,c⁻¹)-sampling, under the condition ofEq. 73:

λ_(+1,−1) ^(A+)=cos Φ; λ_(+1,−1) ^(D+)=0

λ_(+1,−1) ^(A−)=−sin Φ; λ_(+1,−1) ^(D−)=0  (80).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 1},{- 1}}^{D} = 1}{M_{{+ 1},{- 1}}^{Q} = \frac{{\cos \; \Phi}\; }{{{\cos \; \Phi}} + {{\sin \; \Phi}}}}{{M_{{+ 1},{- 1}}^{A} = {{\cos \; \Phi}}},}} & (81)\end{matrix}$

(c₊₃,c⁻³)-Sampling (PMS)

Given Eq. 19, one obtains for (c₊₃,c⁻³)-sampling, under the condition ofEq. 73:

λ_(+3,−3) ^(A+)=cos Φ; λ_(+3,−3) ^(D+)=0

λ_(+3,−3) ^(A−)=sin Φ; λ_(+3,−3) ^(D−)=0  (82).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 3},{- 3}}^{D} = 1}{M_{{+ 3},{- 3}}^{Q} = \frac{{\cos \; \Phi}\; }{{{\cos \; \Phi}} + {{\sin \; \Phi}}}}{{M_{{+ 3},{- 3}}^{A} = {{\cos \; \Phi}}},}} & (83)\end{matrix}$

(c₊₁,c⁻¹,c₊₃,C⁻³)-Sampling (DPMS)

Given Eq. 22, one obtains for (c₊₁,c⁻¹,c₊₃,c⁻³)-sampling, under thecondition of Eq. 73:

λ_(+1,−1,+3,−3) ^(A+)=2 cos Φ; λ_(+1,−1,+3,−3) ^(D+)=0

λ_(+1,−1,+3,−3) ^(A−)=0; λ_(+1,−1,+3,−3) ^(D−)=0  (84).

so that the figures of merit are given by

M _(+1,−1,+3,−3) ^(D)=1

M _(+1,−1,+3,−3) ^(Q)=1

M _(+1,−1,+3,−3) ^(A)=|cos Φ|  (85).

0 and π/2-Shifted Mirrored Sampling

(c₊₀,c⁻²)-Sampling (PMS)

Given Eq. 26, one obtains for (c₊₀,c⁻²)-sampling, under the condition ofEq. 73:

λ_(+0,−2) ^(A+)=cos Φ; λ_(+0,−2) ^(D+)=0

λ_(+0,−2) ^(A−)=0; λ_(+0,−2) ^(D−)=−sin Φ  (86).

so that the figures of merit are given by

$\begin{matrix}{{M_{{+ 0},{- 2}}^{D} = 1}{M_{{+ 0},{- 2}}^{Q} = \frac{{\cos \; \Phi}\; }{{{\cos \; \Phi}} + {{\sin \; \Phi}}}}{{M_{{+ 0},{- 2}}^{A} = {{\cos \; \Phi}}},}} & (87)\end{matrix}$

(c⁻⁰,c₊₂)-Sampling (PMS)

Given Eq. 30, one obtains for (c⁻⁰,c₊₂)-sampling, under the condition ofEq. 73:

λ_(−0,+2) ^(A+)=cos Φ; λ_(−0,+2) ^(D+)=0

λ_(−0,+2) ^(A−)=0; λ_(−0,+2) ^(D−)=−sin Φ  (88).

so that the figures of merit are given by

$\begin{matrix}{{M_{{- 0},{+ 2}}^{D} = 1}{M_{{- 0},{+ 2}}^{Q} = \frac{{\cos \; \Phi}\; }{{{\cos \; \Phi}} + {{\sin \; \Phi}}}}{M_{{- 0},{+ 2}}^{A} = {{{\cos \; \Phi}}.}}} & (89)\end{matrix}$

(c₊₀,c⁻²,c⁻⁰,c₊₂)-Sampling (DPMS)

Given Eq. 33, one obtains for (c₊₀,c⁻², c⁻⁰,c₊₂)-sampling, under thecondition of Eq. 73:

λ_(+0,−2,−0,+2) ^(A+)=2 cos Φ; λ_(+0,−2,−0,+2) ^(D+)=0

λ_(+0,−2,−0,+2) ^(A−)=0; λ_(+0,−2,−0,+2) ^(D−)=0  (90).

so that the figures of merit are given by

M _(+0,−2,−0,+2) ^(D)=1

M _(+0,−2,−0,+2) ^(Q)=1

M _(+0,−2,−0,+2) ^(A)=|cos Φ|  (91).

Extension to Multi-Dimensional NMR

Another aspect of the present invention relates to a general applicationfor Extension to Multi-dimensional NMR.

The sections above introduce different (c_(p),c_(q))-samplings([p,q]=[+0,+2], [−0,−2] for ‘States’ sampling; [+1,−1], [+3,−3] for π/4and 3π/4-shifted mirrored sampling; [+0,−2], [−0,+2] for 0 andπ/2-shifted mirrored sampling), the corresponding interferogram vectorsC_(p,q) of size 2×1 and the D_(p,q) matrices of size 2×2, for a singleindirect dimension. To summarize, the D_(p,q) matrices defined thus farare:

$\begin{matrix}{{{D_{{+ 0},{+ 2}} = {D_{{- 0},{+ 2}} = \begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}};}{D_{{- 0},{- 2}} = {D_{{+ 0},{- 2}} = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}}{{D_{{+ 1},{- 1}} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}};}{D_{{+ 3},{- 3}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}.}}} & (92)\end{matrix}$

For the sake of brevity, we also define

Q=[1 i]  (93),

and its conjugate

Q*=[1 −i]  (94).

The sampling schemes of the above sections, can readily be generalizedto K+1 indirect dimensions of a multi-dimensional NMR experiment, whereK+1 chemical shifts α₀, α₁, . . . α_(K), that are associated with phaseshifts Φ_(±n,0), Φ_(±n,1), . . . Φ_(±n,K), are sampled using the samesampling scheme. For (c_(p),c_(q))-sampling, the correspondinginterferogram vector C_(p,q)(t_(K),t_(K−1), . . . , t₀) of size2^(K+1)×1, the corresponding transformation matrix D_(p,q)(K) of size2^(K+1)×2^(K+1) and the vector Q(K) of size 1×2^(K+1) are obtained byK-fold tensor product formation:

$\begin{matrix}\begin{matrix}{{C_{p,q}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} = {\begin{bmatrix}{c_{p}\left( t_{K} \right)} \\{c_{q}\left( t_{K} \right)}\end{bmatrix} \otimes \begin{bmatrix}{c_{p}\left( t_{K - 1} \right)} \\{c_{q}\left( t_{K - 1} \right)}\end{bmatrix} \otimes \ldots \otimes \begin{bmatrix}{c_{p}\left( t_{1} \right)} \\{c_{q}\left( t_{1} \right)}\end{bmatrix} \otimes}} \\{\begin{bmatrix}{c_{p}\left( t_{0} \right)} \\{c_{q}\left( t_{0} \right)}\end{bmatrix}} \\{= {\overset{K}{\underset{j = 0}{\otimes}}C_{p,q}}} \\{{D_{p,q}(K)} = {\left( D_{p,q} \right)_{K} \otimes \left( D_{p,q} \right)_{K - 1} \otimes \ldots \otimes \left( D_{p,q} \right)_{1} \otimes \left( D_{p,q} \right)_{0}}} \\{= {\overset{K}{\underset{j = 0}{\otimes}}D_{p,q}}} \\{{Q(K)} = {(Q)_{K} \otimes (Q)_{K - 1} \otimes \ldots \otimes (Q)_{1} \otimes (Q)_{0}}} \\{= {\overset{K}{\underset{j = 0}{\otimes}}{Q.}}}\end{matrix} & (95)\end{matrix}$

After FT, the frequency domain peaks of the K+1 dimensional spectrum canbe obtained by an equivalent tensor product formation in the frequencydomain using the description of Eq. 1

$\begin{matrix}\begin{matrix}{{F(K)} = {(F)_{K} \otimes (F)_{K - 1} \otimes \ldots \otimes (F)_{1} \otimes (F)_{0}}} \\{= {\overset{K}{\underset{j = 0}{\otimes}}F}} \\{{\lambda (K)} = {(\lambda)_{K} \otimes (\lambda)_{K - 1} \otimes \ldots \otimes (\lambda)_{1} \otimes (\lambda)_{0}}} \\{= {\overset{K}{\underset{j = 0}{\otimes}}{\lambda.}}}\end{matrix} & (96)\end{matrix}$

In the following, the generalization of mirrored sampling (MS) formultidimensional NMR is derived in two stages. First, for π/4 and3π/4-shifted mirrored sampling of two indirect dimensions (K=1), andsecond, generalization to arbitrary K for all the sampling schemes.

Two Indirect Dimensions (K=1)

(c₊₁,c⁻¹)-Sampling (PMS)

Starting with Eq. 14, the complex signal for two indirect dimensions isproportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 1},{- 1}}\left( {t_{1},t_{0}} \right)} \propto {\underset{j = 0}{\overset{1}{\otimes}}{{QD}_{{+ 1},{- 1}}{C_{{+ 1},1}\left( t_{j} \right)}}}} = {{{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 1}\left( t_{1} \right)} \\{c_{- 1}\left( t_{1} \right)}\end{bmatrix}} \otimes}} \\{{{{\frac{1}{2}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 1}\left( t_{0} \right)} \\{c_{- 1}\left( t_{0} \right)}\end{bmatrix}}} \\{= {\begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},1}} + {\cos \; \Phi_{{- 1},1}}} \right)} \\{^{{{\alpha}\;}_{1}t_{1}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},1}} - {\sin \; \Phi_{{- 1},1}}} \right)} \\{{^{{\pi}\;/2}^{{\alpha}_{1}t_{1}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},1}} + {\sin \; \Phi_{{- 1},1}}} \right)} \\{^{{- {{\alpha}\;}_{1}}t_{1}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},1}} - {\cos \; \Phi_{{- 1},1}}} \right)} \\{^{{\pi}/2}^{{- {\alpha}_{1}}t_{1}}}\end{pmatrix} \otimes}} \\{{\begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}}} \right)} \\{^{{{\alpha}\;}_{0}t_{0}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}}} \right)} \\{{^{{\pi}\;/2}^{{\alpha}_{0}t_{0}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}}} \right)} \\{^{{- {{\alpha}\;}_{0}}t_{0}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}}} \right)} \\{^{{\pi}/2}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}.}}\end{matrix} & (97)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 1},{- 1}}\left( {t_{1,}t_{0}} \right)} \right)} \right)} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \mspace{11mu} \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1\;} +} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {D_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},{0\_}}\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {D_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {A_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right){\left( {D_{0} -} \right).}}}} & (98)\end{matrix}$

The desired peak component is given by (A₁+)(A₀+), which is absorptivein both dimensions. Other terms in Eq. 98 represent mixed phase peakcomponents at the desired or the quad peak location. Under the conditionof identical secondary phase shifts for forward and backward samplings(Eq.35), Eq.98 simplifies to

Re(F_(C)(S_(+1,−1)(t₁,t_(0)))∝(cos Φ) _(1,1) cos Φ_(1,0))(A₁+)(A₀+)−(cosΦ_(1,1), sin Φ_(1,0))(A₁+)(A₀−)−(sin Φ_(1,1) cos Φ_(1,0))(A₁−)(A₀+)+(sinΦ_(1,1) sin Φ_(1,0))(A₁−)(A₀−)  (99),

where, the dispersive peak components are eliminated and only theabsorptive peak components at the desired and the quad positions remain.Under the condition of secondary phase shifts independent of n (Eq. 54),the peak components of Eq.98 remain unchanged. Under the condition ofidentical secondary phase shifts (Eq. 73), Eq. 98 simplifies to

Re(F_(C)(S_(+1,−1)(t₁,t_(0)))∝(cos Φ) ₁ cos Φ₀)(A₁+)(A₀+)−(cos Φ₁ sinΦ₀)(A₁+)(A₀−)−(sin Φ₁ cos Φ₀)(A₁−)(A₀+)+(sin Φ₁ sinΦ₀)(A₁−)(A₀−)  (100).

(c₊₃,c⁻³)-Sampling (PMS)

Starting with Eq. 18, the complex signal for two indirect dimensions isproportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 3},{- 3}}\left( {t_{1},t_{0}} \right)} \propto {\underset{j = 0}{\overset{1}{\otimes}}{{QD}_{{+ 3},{- 3}}{C_{{+ 3},3}\left( t_{j} \right)}}}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}} \\{{\begin{bmatrix}{c_{+ 3}\left( t_{1} \right)} \\{c_{- 3}\left( t_{1} \right)}\end{bmatrix} \otimes}} \\{{{{\frac{1}{2}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 3}\left( t_{0} \right)} \\{c_{- 3}\left( t_{0} \right)}\end{bmatrix}}} \\{= {\begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},1}} + {\cos \; \Phi_{{- 3},1}}} \right)} \\{^{{{\alpha}\; t}\;} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},1}} - {\sin \; \Phi_{{- 3},1}}} \right)} \\{{^{{\pi}\;/2}^{{\alpha}\; t}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},1}} + {\sin \; \Phi_{{- 3},1}}} \right)} \\{^{{- {\alpha}}\; t} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},1}} - {\cos \; \Phi_{{- 3},1}}} \right)} \\{^{{\pi}/2}^{{- {\alpha}}\; t}}\end{pmatrix} \otimes}} \\{{\begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}} \right)} \\{^{{\alpha}\; t} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}} \right)} \\{{^{{\pi}\;/2}^{{\alpha}\; t}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},0}}} \right)} \\{^{{- {\alpha}}\; t} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},0}}} \right)} \\{^{{\pi}/2}^{{- {\alpha}}\; t}}\end{pmatrix}.}}\end{matrix} & (101)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 3},{- 3}}\left( {t_{1,}t_{0}} \right)} \right)} \right)} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \mspace{11mu} \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1\;} +} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{1} -} \right){\left( {D_{0} -} \right).}}}} & (102)\end{matrix}$

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq. 102 simplifies to

Re(F_(C)(S_(+3,−3)(t₁,t_(0)))∝(cos Φ) _(3,1) cos Φ_(3,0))(A₁+)(A₀+)−(cosΦ_(3,1), sin Φ_(3,0))(A₁+)(A₀−)−(sin Φ_(3,1) cos Φ_(3,0))(A₁−)(A₀+)+(sinΦ_(3,1) sin Φ_(3,0))(A₁−)(A₀−)  (103).

Under the condition of secondary phase shifts independent of n (Eq. 54),the peak components of Eq. 102 remain unchanged. Under the condition ofidentical secondary phase shifts (Eq. 73), Eq. 102 simplifies to

Re(F_(C)(S_(+3,−3)(t₁,t_(0)))∝(cos Φ) ₁ cos Φ₀)(A₁+)(A₀+)+(cos Φ₁, sinΦ₀)(A₁+)(A₀−)+(sin Φ₁ cos Φ₀)(A₁−)(A₀+)+(sin Φ₁ sinΦ₀)(A₁−)(A₀−)  (104).

(c₊₁,c⁻¹,c₊₃,C⁻³)-Sampling (DPMS)

Starting with Eq. 21, the complex signal for two indirect dimensions isproportional to

S _(+1,−1,+3,−3)(t ₁ ,t ₀)=└S _(+1,−1)(t ₁)+S ₊3,−3(t ₁)┘

└S _(+1,−1)(t ₀)+S _(+3,−3)(t ₀)┘∝S ₊ _(1,−1)(t ₁)S _(+1,−1)(t ₀)+S_(+1, −1)(t ₁)S _(+3,−3)(t ₀)+S _(+3,−3)(t ₁)S _(+1, −1)(t ₀)+S_(+3,−3)(t ₁)S _(+3,−3)(t ₀)  (105).

DPMS for two indirect dimensions requires all combinations of π/4 and3π/4-shifted mirrored sampling of the time domains. The first and thelast term in Eq. 105 are represented, respectively, by Eqs. 98 and 102.It is then straightforward to show that the signals of the spectrumcorresponding to the second term in Eq. 105 is proportional to

$\begin{matrix}{{{Re}\left( {F_{C}\left( {{S_{{+ 1},{- 1}}\left( t_{1} \right)}{S_{{+ 3},{- 3}}\left( t_{0} \right)}} \right)} \right)} \propto {{\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\sin_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {D_{0} -} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( A_{1} \right)\left( {A_{0} +} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {D_{0} -} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right)\left( {A_{0} +} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right){\left( {D_{0} -} \right).}}}} & (106)\end{matrix}$

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq.106 simplifies to

Re(F_(C)(S_(+1,−1)(t₁)S_(+3,−3)(t_(0)))∝(cos Φ) _(1,1) cosΦ_(3,0))(A₁+)(A₀+)+(cos Φ_(1,1), sin Φ_(3,0))(A₁+)(A₀−)−(sin Φ_(1,1) cosΦ_(3,0))(A₁−)(A₀+)−(sin Φ_(1,1) sin Φ_(3,0))(A₁−)(A₀−)  (107).

Under the condition of secondary phase shifts independent of n (Eq. 54),the peak components of Eq. 106 remain unchanged. Under the condition ofidentical secondary phase shifts (Eq. 73), Eq. 106 simplifies to

Re(F_(C)(S_(+1,−1)(t₁)S_(+3,−3)(t_(0)))∝(cos Φ) ₁ cos Φ₀)(A₁+)(A₀+)+(cosΦ₁ sin Φ₀)(A₁+)(A₀−)−(sin Φ₁ cos Φ₀)(A₁−)(A₀+)−(sin Φ₁ sinΦ₀)(A₁−)(A₀−)  (108).

The signals of the spectrum corresponding to the third term in Eq. 105is proportional to

${{Re}\left( {F_{C}\left( {{S_{{+ 3},{- 3}}\left( t_{1} \right)}{S_{{+ 1},{- 1}}\left( t_{0} \right)}} \right)} \right)} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \mspace{11mu} \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1\;} +} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}\left( {D_{1} -} \right)\left( {D_{0} -} \right)}}$

(109).

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq.109 simplifies to

Re(F_(C)(S_(+1,−1)(t₁)S_(+3,−3)(t_(0)))∝(cos Φ) _(1,1) cosΦ_(3,0))(A₁+)(A₀+)+(cos Φ_(1,1), sin Φ_(3,0))(A₁+)(A₀−)+(sin Φ_(3,1) cosΦ_(1,0))(A₁−)(A₀+)−(sin Φ_(3,1) sin Φ_(1,0))(A₁−)(A₀−)  (110).

Under the condition of secondary phase shifts independent of n (Eq. 54),the peak component pattern of the Eq. 109 does not change. Under thecondition of identical secondary phase shifts (Eq. 73), Eq. 109simplifies to

Re(F_(C)(S_(+3,−3)(t₁)S_(+1,−1)(t_(0)))∝(cos Φ) ₁ cos Φ₀)(A₁+)(A₀+)−(cosΦ₁, sin Φ₀)(A₁+)(A₀−)+(sin Φ₁ cos Φ₀)(A₁−)(A₀+)−(sin Φ_(3,1) sinΦ_(1,0))(A₁−)(A₀−)  (111).

Addition of Eqs.98, 102, 106 and 109 gives the complex signal for(c₊₁,c⁻¹,c₊₃,c⁻³)-sampling for two indirect dimensions and isproportional to

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 1},{- 1},{- 3},{- 3}}\left( {t_{1},t_{0}} \right)} \right)} \right)} \propto {{\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin}\; + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} +} \right)\left( {D_{0} -} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {A_{0} +} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{{- 1},1}\;}\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {D_{0} +} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {A_{0} -} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} +} \right)\left( {D_{0} -} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {A_{0} +} \right)} - {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\left( {{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{+}\cos \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{1} -} \right)\left( {D_{0} -} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right)\left( {A_{0} +} \right)} - {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right)\left( {D_{0} +} \right)} + {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right)\left( {A_{0} -} \right)} + {\frac{1}{4}\left( {{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{1} -} \right){\left( {D_{0} -} \right).}}}} & (112)\end{matrix}$

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq. 112 simplifies to

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 1},{- 1},{+ 3},{- 3}}\left( {t_{1},t_{0}} \right)} \right)} \right)} \propto {{\begin{pmatrix}{{\cos \; \Phi_{1,1}\cos \; \Phi_{1,0}} + {\cos \; \Phi_{1,1}\cos \; \Phi_{3,0}} +} \\{{\cos \; \Phi_{3,1}\cos \; \Phi_{1,0}} + {\cos \; \Phi_{3,1}\cos \; \Phi_{3,0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} +} \right)} - {\begin{pmatrix}{{\cos \; \Phi_{1,1}\sin \; \Phi_{1,0}} - {\cos \; \Phi_{1,1}\sin \; \Phi_{3,0}} +} \\{{\cos \; \Phi_{3,1}\sin \; \Phi_{1,0}} - {\cos \; \Phi_{3,1}\sin \; \Phi_{3,0}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} -} \right)} - {\begin{pmatrix}{{\sin \; \Phi_{1,1}\cos \; \Phi_{1,0}} + {\sin \; \Phi_{1,1}\cos \; \Phi_{3,0}} -} \\{{\sin \; \Phi_{3,1}\cos \; \Phi_{1,0}} - {\sin \; \Phi_{3,1}\cos \; \Phi_{3,0}}}\end{pmatrix}\left( {A_{1} -} \right)\left( {A_{0} +} \right)} + {\begin{pmatrix}{{\sin \; \Phi_{1,1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{1,1}\sin \; \Phi_{3,0}} -} \\{{\sin \; \Phi_{3,1}\sin \; \Phi_{1,0}} + {\sin \; \Phi_{3,1}\sin \; \Phi_{3,0}}}\end{pmatrix}\left( {A_{1} -} \right){\left( {A_{0} -} \right).}}}} & (113)\end{matrix}$

Two dimensional FT reveals absorptive peaks at the desired and the quadpositions. Under the condition of secondary phase shifts independent ofn (Eq. 54), Eq.112 simplifies to

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{+ 1},{- 1},{+ 3},{- 3}}\left( {t_{1},t_{0}} \right)} \right)} \right)} \propto {{\begin{pmatrix}{{\cos \; \Phi_{+ {,1}}\cos \; \Phi_{+ {,0}}} + {\cos \; \Phi_{+ {,1}}\cos \; \Phi_{- {,0}}} +} \\{{\cos \; \Phi_{- {,1}}\cos \; \Phi_{+ {,0}}} + {\cos \; \Phi_{- {,1}}\cos \; \Phi_{- {,0}}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {A_{0} +} \right)} + {\begin{pmatrix}{{\cos \; \Phi_{+ {,1}}\sin \; \Phi_{+ {,0}}} - {\cos \; \Phi_{+ {,1}}\sin \; \Phi_{- {,0}}} +} \\{{\cos \; \Phi_{- {,1}}\sin \; \Phi_{+ {,0}}} - {\cos \; \Phi_{- {,1}}\sin \; \Phi_{- {,0}}}}\end{pmatrix}\left( {A_{1} +} \right)\left( {D_{0} +} \right)} + {\begin{pmatrix}{{\sin \; \Phi_{+ {,1}}\cos \; \Phi_{+ {,0}}} + {\sin \; \Phi_{+ {,1}}\cos \; \Phi_{- {,0}}} -} \\{{\sin \; \Phi_{- {,1}}\cos \; \Phi_{+ {,0}}} - {\sin \; \Phi_{- {,1}}\cos \; \Phi_{- {,0}}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {A_{0} +} \right)} + {\begin{pmatrix}{{\sin \; \Phi_{+ {,1}}\sin \; \Phi_{+ {,0}}} - {\sin \; \Phi_{+ {,1}}\sin \; \Phi_{- {,0}}} -} \\{{\sin \; \Phi_{- {,1}}\sin \; \Phi_{+ {,0}}} + {\sin \; \Phi_{- {,1}}\sin \; \Phi_{- {,0}}}}\end{pmatrix}\left( {D_{1} +} \right)\left( {D_{0} +} \right)}}},} & (114)\end{matrix}$

Two dimensional FT reveals a single mixed phase peak at the desiredposition. This proves that, imbalance between forward and backwardsamplings can not eliminate dispersive components. Under the conditionof identical secondary phase shifts (Eq. 73), Eq. 112 simplifies to

Re(F_(C)(S_(+1,−1,+3,−3)(t₁,t₀)))∝4 cos Φ₁ cos Φ₀(A₁+)(A₀+)  (115).

Two dimensional FT reveals a single peak, 4 cos Φ₁ cos Φ₀ (A₁+)(A₀+),which is purely absorptive in both dimensions.

Arbitrary Number of Indirect Dimensions

The complex time domain signal results in a sum over products of allpossible permutations of cos Φ±n,j and sin Φ_(±n,j).

(c₊₀,c₊₂)-Sampling

Starting with Eq. 3, the complex signal for multiple States quadraturedetection is proportional to

$\begin{matrix}{{{S_{{+ 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\overset{K}{\underset{j = 0}{\otimes}}{{QD}_{{+ 0},{+ 2}}{C_{{+ 0},{+ 2}}\left( t_{j} \right)}}}} = {{{{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\left\lbrack \begin{matrix}{c_{+ 0}\left( t_{K} \right)} \\{c_{+ 2}\left( t_{K} \right)}\end{matrix} \right\rbrack} \otimes \begin{bmatrix}1 & \end{bmatrix}}{\quad{{{\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack \begin{bmatrix}{c_{+ 0}\left( t_{K - 1} \right)} \\{c_{+ 2}\left( t_{K - 1} \right)}\end{bmatrix}} \otimes \ldots \otimes {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\begin{bmatrix}{c_{+ 0}\left( t_{0} \right)} \\{c_{+ 2}\left( t_{0} \right)}\end{bmatrix}}} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{+ 2},K}}} \right)^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{+ 2},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)^{\; \alpha_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)^{{- {\alpha}_{K - 1}}t_{K - 1}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{K - 1}}t_{K - 1}}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{+ 2},0}}} \right)^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{+ 2},0}}} \right)^{{- {\alpha}_{0}}t_{0}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}.}}}}}} & (116)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{+ 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{+ 2},K}}} \right)\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{+ 2},K}}} \right)\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{+ 2},K}}} \right)\left( {A_{K} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{+ 2},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {A_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {A_{K - 1} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {D_{K - 1} - 1} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{+ 2},0}}} \right)\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{+ 2},0}}} \right)\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{+ 2},0}}} \right)\left( {A_{0} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (117)\end{matrix}$

The term (A_(K)+)(A_(K−1)+) . . . (A₀+) and the term (D_(K)+)(D_(K−1)+). . . (D₀+), respectively, represent an entirely absorptive and entirelydispersive components of the desired peak located at (α₀,α₁, . . . ,α_(K)), (ii) the term (A_(K)−)(A_(K−1)−) . . . (A₀−) and the term(D_(K)−)(D_(K−1)−) . . . (D₀−), respectively, represent an entirelyabsorptive and entirely dispersive components of the peak located at thequadrature position along all dimensions (−α₀,−α₁, . . . , −α_(K)). Allother terms represent mixed phase peak components at either the desiredor any one of the quad positions.

(c−₀, c⁻²)-Sampling

Starting with Eq. 7, the complex signal for multiple backward Statesquadrature detection is proportional to

$\begin{matrix}{{{S_{{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\overset{K}{\underset{j = 0}{\otimes}}{{QD}_{{- 0},{- 2}}{C_{{- 0},{- 2}}\left( t_{j} \right)}}}} = {{{{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{c_{- 0}\left( t_{K} \right)} \\{c_{- 2}\left( t_{K} \right)}\end{bmatrix}} \otimes \begin{bmatrix}1 & \end{bmatrix}}{\quad{{{\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}\begin{bmatrix}{c_{- 0}\left( t_{K - 1} \right)} \\{c_{- 2}\left( t_{K - 1} \right)}\end{bmatrix}} \otimes \ldots \otimes {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{c_{- 0}\left( t_{0} \right)} \\{c_{- 2}\left( t_{0} \right)}\end{bmatrix}}} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)^{{\alpha}_{K}t_{K}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{K}}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}} \right)^{{\alpha}_{K\; - 1}t_{K - 1}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}} \right)^{{- }\; \alpha_{K - 1}t_{K - 1}}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)^{{\alpha}_{0}t_{0}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{- 2},0}}} \right)^{{- {\alpha}_{0}}t_{0}}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}.}}}}}} & (118)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)\left( {A_{K} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)\left( {A_{K} -} \right)} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}} \right)\left( {A_{K - 1} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}} \right)\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}} \right)\left( {A_{K - 1} -} \right)} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}} \right)\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)\left( {A_{0} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{- 2},0}}} \right)\left( {A_{0} - 1} \right)} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (119)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

(c₊₀,c₊₂,c⁻⁰,c⁻²)-Sampling

Starting with Eq. 10 the complex signal for multiple dual Statesquadrature detection is proportional to

$\begin{matrix}{{{S_{{+ 0},{+ 2},{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{+ 2},K}} + {\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)^{{\alpha}_{K}t_{k}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{+ 2},K}} - {\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{- 2},K}} + {\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{- 2},K}} - {\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{{- }\; \alpha_{K}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{2,{K - 1}}}}\end{pmatrix}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}^{{- {\alpha}_{K - 1}}t_{K - 1}}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; \alpha_{K - 1}t_{K - 1}}}\end{pmatrix} \otimes \ldots \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{+ 2},0}} + {\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)^{\; \alpha_{0}t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{+ 2},0}} - {\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{+ 2},0}} + {\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{- 2},0}}} \right)^{{- {\alpha}_{0}}t_{0}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{+ 2},0}} - {\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}}},} & (120)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 0},{+ 2},{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{+ 2},K}} + {\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{+ 2},K}} - {\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{+ 2},K}} + {\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)\left( {A_{K} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{+ 2},K}} - {\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}\left( {A_{K - 1} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{+ 2},0}} + {\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{+ 2},0}} - {\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{+ 2},0}} + {\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{- 2},0}}} \right)\left( {A_{0} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{+ 2},0}} - {\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (121)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

(c₊₁,c⁻¹)-Sampling (PMS)

Starting with Eq. 14, the complex signal for (c₊₁,c⁻¹)-PMS isproportional to

$\begin{matrix}{{{S_{{+ 1},{- 1}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\overset{K}{\underset{j = 0}{\otimes}}{{QD}_{{+ 1},{- 1}}{C_{{+ 1},{- 1}}\left( t_{j} \right)}}}} = {{{{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 1}\left( t_{K} \right)} \\{c_{- 1}\left( t_{K} \right)}\end{bmatrix}} \otimes {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}}{\quad{{\begin{bmatrix}{c_{+ 1}\left( t_{K - 1} \right)} \\{c_{- 1}\left( t_{K - 1} \right)}\end{bmatrix} \otimes \ldots \otimes {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 1}\left( t_{0} \right)} \\{c_{- 1}\left( t_{0} \right)}\end{bmatrix}}} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} + {\cos \; \Phi_{{- 1},K}}} \right)^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} - {\sin \; \Phi_{{- 1},K}}} \right)^{\; {\pi/2}}^{{\alpha}_{K}t_{K}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} + {\sin \; \Phi_{{- 1},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} - {\cos \; \Phi_{{- 1},K}}} \right)^{{\pi}/2}^{\; \alpha_{K}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},{K - 1}}} + {\cos \; \Phi_{{- 1},{K - 1}}}} \right)^{\; \alpha_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}}} \right) \times^{{\pi}/2}^{{\alpha}_{K - 1}t_{K - 1}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}}} \right)^{{- }\; \alpha_{K - 1}t_{K - 1}}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},{K - 1}}} - {\cos \; \Phi_{{- 1},{K - 1}}}} \right)^{\; {\pi/2}}^{{- {\alpha}_{K - 1}}t_{K - 1}}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}}} \right)^{\; {\pi/2}}^{\; \alpha_{0}t_{0}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},0}}} \right)^{{- }\; \alpha_{0}t_{0}}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},0}}} \right)^{{\pi}/2}^{{- }\; \alpha_{0}t_{0}}}\end{pmatrix}.}}}}}} & (122)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{+ 1},{- 1}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} + {\cos \; \Phi_{{- 1},K}}} \right)\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} - {\sin \; \Phi_{{- 1},K}}} \right)\left( {D_{K} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} + {\sin \; \Phi_{{- 1},K}}} \right)\left( {A_{K} -} \right)} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} - {\cos \; \Phi_{{- 1},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},{K - 1}}} + {\cos \; \Phi_{{- 1},{K - 1}}}} \right)\left( {A_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}}} \right)\left( {D_{K - 1} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}}} \right)\left( {A_{K - 1} -} \right)} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},{K - 1}}} - {\cos \; \Phi_{{- 1},{K - 1}}}} \right)\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}}} \right)\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}}} \right)\left( {D_{0} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}}} \right)\left( {A_{0} -} \right)} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (123)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

(c₊₃,c⁻³)-Sampling (PMS)

Starting with Eq. 18, the complex signal for (c₊₃,c⁻³)-PMS isproportional to

$\begin{matrix}{{\left. {S_{{+ 3},{- 3}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right) \propto {\overset{K}{\underset{j = 0}{\otimes}}{{QD}_{{+ 3},{- 3}}{C_{{+ 3},{- 3}}\left( t_{j} \right)}}}} = {{{{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 3}\left( t_{K} \right)} \\{c_{- 3}\left( t_{K} \right)}\end{bmatrix}} \otimes {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}}{\quad{{\begin{bmatrix}{c_{+ 3}\left( t_{K - 1} \right)} \\{c_{- 3}\left( t_{K - 1} \right)}\end{bmatrix} \otimes \ldots \otimes {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 3}\left( t_{0} \right)} \\{c_{- 3}\left( t_{0} \right)}\end{bmatrix}}} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}} \right)^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}} \right)^{\; {\pi/2}}^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},K}} + {\sin \; \Phi_{{- 3},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},K}} - {\cos \; \Phi_{{- 3},K}}} \right)^{{\pi}/2}^{\; \alpha_{K}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}} \right)^{\; \alpha_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},{K - 1}}}} \right) \times^{{\pi}/2}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},{K - 1}}} + {\sin \; \Phi_{{- 3},{K - 1}}}} \right)^{{- }\; \alpha_{K - 1}t_{K - 1}}} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},{K - 1}}} - {\cos \; \Phi_{{- 3},{K - 1}}}} \right)^{\; {\pi/2}}^{{- {\alpha}_{K - 1}}t_{K - 1}}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}} \right)^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}} \right)^{\; {\pi/2}}^{\; \alpha_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},0}}} \right)^{{- }\; \alpha_{0}t_{0}}} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},0}}} \right)^{{\pi}/2}^{{- }\; \alpha_{0}t_{0}}}\end{pmatrix}.}}}}}} & (124)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{+ 3},{- 3}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}} \right)\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}} \right)\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},K}} + {\sin \; \Phi_{{- 3},K}}} \right)\left( {A_{K} -} \right)} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},K}} - {\cos \; \Phi_{{- 3},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}} \right)\left( {A_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},{K - 1}}}} \right)\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},{K - 1}}} + {\sin \; \Phi_{{- 3},{K - 1}}}} \right)\left( {A_{K - 1} -} \right)} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},{K - 1}}} - {\cos \; \Phi_{{- 3},{K - 1}}}} \right)\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}} \right)\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}} \right)\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},0}}} \right)\left( {A_{0} -} \right)} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (125)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

(c₊₁,c⁻¹,c₊₃,c⁻³)-Sampling (DPMS)

Starting with Eq. 21, the complex signal for (c⁻¹,c⁻¹,c₊₃,c⁻³)-DPMS isproportional to

$\begin{matrix}{{S_{{+ 1},{- 1},{+ 3},{- 3}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} + {\cos \; \Phi_{{- 1},K}} + {\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}} \right)^{{\alpha}_{\;_{K}t_{K}}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} - {\sin \; \Phi_{{- 1},K}} + {\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{\;_{K}t_{K}}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} + {\sin \; \Phi_{{- 1},K}} - {\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} - {\cos \; \Phi_{{- 1},K}} - {\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}} \right)^{\frac{\pi}{2}}^{{- }\; \alpha_{K}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},{K - 1}}} + {\cos \; \Phi_{{- 1},{K - 1}}} +} \\{{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}^{\; \alpha_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}} +} \\{{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}^{\frac{\pi}{2}}^{\; \alpha_{K - 1}t_{K - 1}}} -} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}} -} \\{{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},K}}}\end{pmatrix}^{{- }\; \alpha_{K - 1}t_{K - 1}t}} -} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},{K - 1}}} - {\cos \; \Phi_{{- 1},{K - 1}}} -} \\{{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; \alpha_{K - 1}t_{K - 1}t}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}} + {\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}} \right)^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}} + {\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}} \right)^{\frac{\pi}{2}}^{\; \alpha_{0}t_{0}}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}} - {\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}} \right)^{{- }\; \alpha_{0}t_{0}}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}} - {\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}} \right)^{\frac{\pi}{2}}^{{- }\; \alpha_{0}t_{0}}}\end{pmatrix}.}}} & (126)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 0},{+ 2},{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},K}} + {\cos \; \Phi_{{- 1},K}} +} \\{{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}}\end{pmatrix}\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},K}} - {\sin \; \Phi_{{- 1},K}} +} \\{{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}}\end{pmatrix}\left( {D_{K} +} \right)} -} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},K}} + {\sin \; \Phi_{{- 1},K}} -} \\{{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}}\end{pmatrix}\left( {A_{K} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},K}} - {\cos \; \Phi_{{- 1},K}} -} \\{{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}}\end{pmatrix}\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},{K - 1}}} + {\cos \; \Phi_{{- 1},{K - 1}}} +} \\{{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}\left( {A_{K - 1} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}} +} \\{{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}\left( {D_{K - 1} +} \right)} -} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}} -} \\{{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},K}}}\end{pmatrix}\left( {A_{K - 1} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},{K - 1}}} - {\cos \; \Phi_{{- 1},{K - 1}}} -} \\{{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{0} +} \right)} -} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {A_{0} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}}\end{pmatrix}\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (127)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

(c₊₀,c⁻²)-Sampling (PMS)

Starting with Eq. 25, the complex signal for (c₊₀,c⁻²)-PMS isproportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\underset{j = 0}{\overset{K}{\otimes}}{{QD}_{{+ 0},{- 2}}{C_{{+ 0},{- 2}}\left( t_{j} \right)}}}} = {\left\lbrack \begin{matrix}1 & \end{matrix} \right\rbrack\left\lbrack \begin{matrix}1 & 0 \\0 & 1\end{matrix} \right\rbrack}} \\{{\left\lbrack \begin{matrix}{c_{+ 0}\left( t_{K} \right)} \\{c_{- 2}\left( t_{K} \right)}\end{matrix} \right\rbrack \otimes}} \\{{\left\lbrack \begin{matrix}1 & \end{matrix} \right\rbrack\left\lbrack \begin{matrix}1 & 0 \\0 & 1\end{matrix} \right\rbrack}} \\{{\left\lbrack \begin{matrix}{c_{+ 0}\left( t_{K - 1} \right)} \\{c_{- 2}\left( t_{K - 1} \right)}\end{matrix} \right\rbrack \otimes \ldots \otimes}} \\{{{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 0}\left( t_{0} \right)} \\{c_{- 2}\left( t_{0} \right)}\end{bmatrix}}} \\{= {\begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} +} \\{\cos \; \Phi_{{- 2},K}}\end{pmatrix}} \\{^{{\alpha}_{K}t_{K}} +} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} -} \\{\sin \; \Phi_{{- 2},K}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} -} \\{\cos \; \Phi_{{- 2},K}}\end{pmatrix}} \\{^{{- {\alpha}_{K}}t_{K}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} +} \\{\sin \; \Phi_{{- 2},K}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- {\alpha}_{K}}t_{K}}}\end{pmatrix} \otimes}} \\{\begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} +} \\{\cos \; \Phi_{{- 2},{K - 1}}}\end{pmatrix}} \\{^{{\alpha}_{K - 1}t_{K - 1}} +} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} -} \\{\sin \; \Phi_{{- 2},{K - 1}}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} -} \\{\cos \; \Phi_{{- 2},{K - 1}}}\end{pmatrix}} \\{^{{\alpha}_{K - 1}t_{K - 1}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} +} \\{\sin \; \Phi_{{- 2},{K - 1}}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- {\alpha}_{K - 1}}t_{K - 1}}}\end{pmatrix}} \\{{\otimes \; {\ldots \; \otimes {\begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} +} \\{\cos \; \Phi_{{- 2},0}}\end{pmatrix}} \\{^{{\alpha}_{0}t_{0}} +} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} -} \\{\sin \; \Phi_{{- 2},0}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{{\alpha}_{0}t_{0}}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} -} \\{\cos \; \Phi_{{- 2},0}}\end{pmatrix}} \\{^{{- {\alpha}_{0}}t_{0}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} +} \\{\sin \; \Phi_{{- 2},0}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}.}}}}\end{matrix} & (128)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{+ 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)\left( {A_{K} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}} \right)\left( {A_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}} \right)\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}} \right)\left( {A_{K - 1} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}} \right)\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)\left( {A_{0} -} \right)} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (129)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

(c⁻⁰,c₊₂)-Sampling (PMS)

Starting with Eq. 29, the complex signal for (c⁻⁰,c₊₂)-PMS isproportional to

$\begin{matrix}\begin{matrix}{{{S_{{- 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\underset{j = 0}{\overset{K}{\otimes}}{{QD}_{{- 0},{+ 2}}{C_{{- 0},{+ 2}}\left( t_{j} \right)}}}} = {\begin{bmatrix}1 & i\end{bmatrix}\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack}} \\{{\left\lbrack \begin{matrix}{c_{- 0}\left( t_{K} \right)} \\{c_{+ 2}\left( t_{K} \right)}\end{matrix} \right\rbrack \otimes {\begin{bmatrix}1 & i\end{bmatrix}\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack}}} \\{{\left\lbrack \begin{matrix}{c_{- 0}\left( t_{K - 1} \right)} \\{c_{+ 2}\left( t_{K - 1} \right)}\end{matrix} \right\rbrack \otimes \ldots \otimes}} \\{\begin{bmatrix}1 & i\end{bmatrix}} \\{{\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}\begin{bmatrix}{c_{- 0}\left( t_{0} \right)} \\{c_{+ 2}\left( t_{0} \right)}\end{bmatrix}}} \\{= {\begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{- 0},K}} +} \\{\cos \; \Phi_{{+ 2},K}}\end{pmatrix}} \\{^{{\alpha}_{K}t_{K}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{- 0},K}} -} \\{\sin \; \Phi_{{+ 2},K}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{- 0},K}} -} \\{\cos \; \Phi_{{+ 2},K}}\end{pmatrix}} \\{^{{\alpha}_{K}t_{K}} +} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{- 0},K}} -} \\{\sin \; \Phi_{{+ 2},K}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- {\alpha}_{K}}t_{K}}}\end{pmatrix} \otimes}} \\{\begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{- 0},{K - 1}}} +} \\{\cos \; \Phi_{{+ 2},{K - 1}}}\end{pmatrix}} \\{^{{\alpha}_{K - 1}t_{K - 1}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{- 0},{K - 1}}} -} \\{\sin \; \Phi_{{+ 2},{K - 1}}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{- 0},{K - 1}}} -} \\{\cos \; \Phi_{{+ 2},{K - 1}}}\end{pmatrix}} \\{^{{- {\alpha}_{K - 1}}t_{K - 1}} +} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{- 0},{K - 1}}} +} \\{\sin \; \Phi_{{+ 2},{K - 1}}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- {\alpha}_{K - 1}}t_{K - 1}}}\end{pmatrix}} \\{{\otimes {\ldots \otimes {\begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{- 0},0}} +} \\{\cos \; \Phi_{{+ 2},0}}\end{pmatrix}} \\{^{{\alpha}_{0}t_{0}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{- 0},0}} -} \\{\sin \; \Phi_{{+ 2},0}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{{\alpha}_{0}t_{0}}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{- 0},0}} -} \\{\cos \; \Phi_{{+ 2},0}}\end{pmatrix}} \\{^{{- {\alpha}_{0}}t_{0}} +} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{- 0},0}} +} \\{\sin \; \Phi_{{+ 2},0}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}.}}}}\end{matrix} & (130)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{{Re}\left( {F_{C}\left( {S_{{- 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{+ 2},K}}} \right)\left( {A_{K} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{+ 2},K}}} \right)\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{+ 2},K}}} \right)\left( {A_{K} -} \right)} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{+ 2},K}}} \right)\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {A_{K - 1} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {A_{K - 1} -} \right)} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \; \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{+ 2},0}}} \right)\left( {A_{0} +} \right)} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{+ 2},0}}} \right)\left( {A_{0} -} \right)} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{+ 2},0}}} \right)\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (131)\end{matrix}$

Multidimensional FT reveals a peak component pattern comparable to theone described for Eq. 117.

(c₊₀,c⁻²,c⁻⁰,c₊₂)-Sampling (DPMS)

Starting with Eq. 32, the complex signal for (c₊₀,c⁻²,c⁻⁰,c₊₂)-DPMS isproportional to

$\begin{matrix}{{S_{{+ 0},{+ 2},{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{- 2},K}} + {\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{+ 2},K}}} \right)^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{- 2},K}} - {\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{K}t_{K}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{- 2},K}} + {\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{+ 2},K}}} \right)^{{- {\alpha}_{K}}t_{K}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{- 2},K}} - {\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{K}}t_{K}}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}^{{\alpha}_{K - 1}t_{K - 1}}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}^{\frac{\pi}{2}}^{{\alpha}_{K - 1}t_{K - 1}}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}} + {\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{+ 2},0}}} \right)^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{- 2},0}} - {\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{{\alpha}_{0}t_{0}}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{- 2},0}} + {\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{+ 2},K}}} \right)^{{- {\alpha}_{0}}t_{0}}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{- 2},0}} - {\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{{- {\alpha}_{0}}t_{0}}}\end{pmatrix}.}}} & (132)\end{matrix}$

Multidimensional FT reveals the peak components as

$\begin{matrix}{{{Re}\left( {F_{C}\left( {S_{{+ 0},{+ 2},{- 0},{- 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} \right)} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{- 2},K}} +} \\{{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{+ 2},K}}}\end{pmatrix}\left( {A_{K} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{- 2},K}} -} \\{{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{+ 2},K}}}\end{pmatrix}\left( {D_{K} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{- 2},K}} +} \\{{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{+ 2},K}}}\end{pmatrix}\left( {A_{K} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{- 2},K}} -} \\{{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{+ 2},K}}}\end{pmatrix}\left( {D_{K} -} \right)}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}\left( {A_{K - 1} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}\left( {D_{K - 1} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}\left( {A_{K - 1} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}\left( {D_{K - 1} -} \right)}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}} +} \\{{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{+ 2},0}}}\end{pmatrix}\left( {A_{0} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{- 2},0}} -} \\{{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{+ 2},0}}}\end{pmatrix}\left( {D_{0} +} \right)} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{- 2},0}} +} \\{{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{+ 2},K}}}\end{pmatrix}\left( {A_{0} -} \right)} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{- 2},0}} -} \\{{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{+ 2},0}}}\end{pmatrix}\left( {D_{0} -} \right)}\end{pmatrix}.}}} & (133)\end{matrix}$

The peak component pattern is comparable to the one described for Eq.117.

Extension to G-Matrix FT NMR

The sampling schemes introduced above can be applied to an arbitrarysub-set of K+1 chemical shift evolution periods which are jointlysampled in G-matrix FT (GFT) NMR (Kim et al., J. Am. Chem. Soc.125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004);Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al.,J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol.394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A.102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692(2007), which are hereby incorporated by reference in their entirety),where the K+1 chemical shifts α₀, α₁, . . . α_(K) are associated withphase shifts Φ_(±n,0),Φ_(±n,1), . . . ) Φ_(±n,K) depending on thesampling schemes chosen and measured as linear combinations α₀±κ₁α₁ ± .. . ±κ_(K)α_(K) edited into 2^(K) sub-spectra. The scaling factors κ_(j)enable one to achieve different maximal evolution times for thedifferent jointly sampled shifts (see below).

The complex signal corresponding to the 2^(K) edited sub-spectraobtained after G matrix transformation can be written as

$\begin{matrix}{{{T_{{+ 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} = {{G(K)}{D_{{+ 0},{+ 2}}(K)}{C_{{+ 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)}}}{{where},}} & (134) \\\begin{matrix}\begin{matrix}{{G(K)} = {\begin{bmatrix}1 &  \\1 & {- i}\end{bmatrix}_{K} \otimes \begin{bmatrix}1 &  \\1 & {- i}\end{bmatrix}_{K - 1} \otimes \ldots \otimes \begin{bmatrix}1 &  \\1 & {- i}\end{bmatrix}_{1} \otimes \begin{bmatrix}1 & \end{bmatrix}_{0}}} \\{= {\begin{bmatrix}Q \\Q^{*}\end{bmatrix}_{K} \otimes \begin{bmatrix}Q \\Q^{*}\end{bmatrix}_{K - 1} \otimes \ldots \otimes \begin{bmatrix}Q \\Q^{*}\end{bmatrix}_{1} \otimes \lbrack Q\rbrack_{0}}}\end{matrix} \\\begin{matrix}{{D_{{+ 0},{+ 2}}(K)} = {\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}_{K} \otimes \begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}_{K - 1} \otimes \ldots \otimes \begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}_{1} \otimes}} \\{\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}_{0}}\end{matrix} \\\begin{matrix}{{C_{{+ 0},{+ 2}}\left( {t_{K},\ldots \mspace{14mu},t_{0}} \right)} = {\begin{bmatrix}{c_{+ 0}\left( t_{K} \right)} \\{c_{+ 2}\left( t_{K} \right)}\end{bmatrix} \otimes \begin{bmatrix}{c_{+ 0}\left( t_{K - 1} \right)} \\{c_{+ 2}\left( t_{K - 1} \right)}\end{bmatrix} \otimes \ldots \otimes}} \\{{\begin{bmatrix}{c_{+ 0}\left( t_{1} \right)} \\{c_{+ 2}\left( t_{1} \right)}\end{bmatrix} \otimes {\begin{bmatrix}{c_{+ 0}\left( t_{0} \right)} \\{c_{+ 2}\left( t_{0} \right)}\end{bmatrix}.}}}\end{matrix}\end{matrix} & \;\end{matrix}$

The time domain signal S_(+0,+2)(t) given in Eq. 3 encodes +α:

$\begin{matrix}{{{{S_{{+ 0},{+ 2}}(t)} \propto {{QD}_{{+ 0},{+ 2}}{C_{{+ 0},{+ 2}}(t)}}} = {{{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\begin{bmatrix}{c_{+ 0}(t)} \\{c_{+ 2}(t)}\end{bmatrix}}\mspace{301mu} = {{\begin{bmatrix}1 & {- }\end{bmatrix}\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \Phi} \right)} \\{- {\sin \left( {{\alpha \; t} + \Phi} \right)}}\end{bmatrix}}\mspace{301mu} = {{\cos \; \Phi \; ^{{\alpha}\; t}} + {\sin \; {\Phi }^{\frac{\pi}{2}}^{{\alpha}\; t}}}}}},} & (135)\end{matrix}$

The complex conjugate of S_(+0,+2)(t), denoted as S*_(+0,+2) (t),encodes −α:

$\begin{matrix}{{{S_{{+ 0},{- 2}}^{*}(t)} \propto {Q^{*}D_{{+ 0},{+ 2}}{C_{{+ 0},{+ 2}}( t)}} \propto {{\left\lbrack \begin{matrix}{ 1} & {- i}\end{matrix} \right\rbrack\left\lbrack \begin{matrix}1 & 0 \\0 & {- 1}\end{matrix} \right\rbrack}\left\lbrack \begin{matrix}{c_{- 0}(t)} \\{c_{+ 2}(t)}\end{matrix} \right\rbrack}} = {{\left\lbrack \begin{matrix}1 & \end{matrix} \right\rbrack\left\lbrack \begin{matrix}{\cos \left( {{{+ \alpha}\; t} + \Phi} \right)} \\{- {\sin \left( {{\alpha \; t} + \Phi} \right)}}\end{matrix} \right\rbrack} = {{\cos \; {\Phi }^{{- }\; \alpha \; t}} - {\sin \; {\Phi }^{\; \frac{\pi}{2}}{^{{- }\; \alpha \; t}.}}}}} & (136)\end{matrix}$

In GFT NMR, t is coupled to the increment of all jointly sampledevolution periods t_(K) by

t=t ₀ =t ₁/κ₁ =t ₂/κ₂ = . . . =t _(j)/κ_(j) = . . . =t_(K)/κ_(K)  (137),

that is, t₁ . . . t_(K) are scaled with respect t₀. Hence, for thej^(th) chemical shift α_(j), one has that t=t_(j)/κ_(j).

With Eqs. 134-137, the vector T_(+0,+2) can be written as

$\begin{matrix}{{T_{{+ 0},{- 2}}(t)} = {{\begin{bmatrix}{S_{{- 0},{+ 2}}(t)} \\{S_{{- 0},{+ 2}}^{*}(t)}\end{bmatrix}_{K} \otimes \begin{bmatrix}{S_{{+ 0},{- 2}}(t)} \\{S_{{+ 0},{- 2}}^{*}(t)}\end{bmatrix}_{K - 1} \otimes \ldots \otimes \begin{bmatrix}{S_{{+ 0},{+ 2}}(t)} \\{S_{{+ 0},{+ 2}}^{*}(t)}\end{bmatrix}_{1} \otimes \left\lbrack {S_{{+ 0},{+ 2}}(t)} \right\rbrack_{0}} \propto {\begin{bmatrix}^{{\alpha}_{K}t} \\^{{- {\alpha}_{K}}t}\end{bmatrix} \otimes \begin{bmatrix}^{{\alpha}_{K - 1}t} \\^{{- {\alpha}_{K - 1}}t}\end{bmatrix} \otimes \ldots \otimes \begin{bmatrix}^{{\alpha}_{1}t} \\^{{- {\alpha}_{1}}t}\end{bmatrix} \otimes {^{{\alpha}_{0}t}.}}}} & (138)\end{matrix}$

A specific element of T_(+0,+2) (t), denoted here as T_(+0,+2) (t,M),represents a sub-spectrum in which a particular linear combination ofthe jointly sampled chemical shifts is measured, that is, one linearcombination out of the set {α₀±κ₁α₁± . . . ± κ_(K)α_(K)} is selected.This particular linear combination can be identified with a sign vectorM=[M_(K)M_(K−1) . . . M₀], where M_(j)=1 for +α_(j) or M_(j)=−1 for−α_(j), so that

$\begin{matrix}{{{T_{{+ 0},{+ 2}}\left( {t,M} \right)} \propto {^{\; M_{K}\alpha_{K}t} \otimes ^{\; M_{K - 1}\alpha_{K - 1}t} \otimes \ldots \otimes ^{\; M_{0}\alpha_{0}t}}} = {\underset{j = 0}{\overset{K}{\otimes}}{^{\; M_{j}\alpha_{j}t}.}}} & (139)\end{matrix}$

Given M, S_(+0,±2) (t) defined in Eq. 135 and encoding +α_(j), andS*_(+0,+2) (t) defined in Eq. 136 and encoding −α_(j), results in

$\begin{matrix}{{T_{{+ 0},{+ 2}}\left( {t,M} \right)} \propto {\underset{j = 0}{\overset{K}{\otimes}}{\left( {{\cos \; \Phi_{j}^{\; M_{j}\alpha_{j}t}} + {M_{j}\sin \; \Phi_{j}^{\frac{\pi}{2}}^{\; M_{j}\alpha_{j}t}}} \right).}}} & (140)\end{matrix}$

For (c_(p),c_(q))-sampling, the time domain signal for the sub-spectrum,T_(p,q)(t, M), can be obtained using the respective complex time domainsignal S_(p,q)(t) given in the section above and the correspondingconjugate S*_(p,q) (t).

In the following, the generalization of mirrored sampling (MS) in GFTprojection NMR is derived in two steps. First, for π/4 and 3π/4-shiftedmirrored sampling of two jointly sampled chemical shifts (K=1) andsecond for arbitrary K for all sampling schemes.

Two Jointly Sampled Chemical Shifts (K=1)

For two jointly sampled chemical shifts, one has two edited sub-spectra,corresponding to M=[1 1], where the sum of the chemical shifts isrecorded, and to M=[−1 1], where the difference of the chemical shiftsis recorded. The following derivations consider only the sub-spectrumM=[1 1]. The derivations for the other sub-spectrum are analogous.

(c₊₁,c⁻¹)-Sampling (PMS)

With S+_(1,−1)(t) of Eq. 14 the complex conjugate is proportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 1},{- 2}}^{*}(t)} \propto {\begin{bmatrix}1 & {- i}\end{bmatrix}D_{{+ 1},{- 1}}{C_{{+ 1},{- 1}}(t)}}} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & {- }\end{bmatrix}}} \\{{\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}\begin{bmatrix}{c_{+ 1}(t)} \\{c_{- 1}(t)}\end{bmatrix}}} \\{= {\frac{1}{\sqrt{2}}\begin{bmatrix}{1 + } & {1 - }\end{bmatrix}}} \\{\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 1}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 1}} \right)}\end{bmatrix}} \\{= {\frac{1}{\sqrt{2}}\begin{pmatrix}{\begin{pmatrix}{{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} +} \\{\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix} +} \\{\begin{pmatrix}{{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} -} \\{\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix}}\end{pmatrix}}} \\{{{\cos \left( {\alpha \; t} \right)} -}} \\{{\frac{1}{\sqrt{2}}\begin{pmatrix}{\begin{pmatrix}{{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} -} \\{\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix} +} \\{\begin{pmatrix}{{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} +} \\{\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix}}\end{pmatrix}}} \\{{\sin \left( {\alpha \; t} \right)}} \\{= {\frac{1}{\sqrt{2}} {\quad \begin{pmatrix}{\begin{pmatrix}{{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} +} \\{\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix} +} \\{\begin{pmatrix}{{\cos \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} -} \\{\cos \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix}}\end{pmatrix}}}} \\{{\frac{^{{\alpha}\; t} + ^{{- {\alpha}}\; t}}{2} -}} \\{{\frac{1}{\sqrt{2}}\begin{pmatrix}{\begin{pmatrix}{{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} -} \\{\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix} +} \\{\begin{pmatrix}{{\sin \left( {\frac{\pi}{4} + \Phi_{+ 1}} \right)} +} \\{\sin \left( {\frac{\pi}{4} + \Phi_{- 1}} \right)}\end{pmatrix}}\end{pmatrix}}} \\{\frac{^{{\alpha}\; t} - ^{{- {\alpha}}\; t}}{2}} \\{= {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} +} \\{{\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}} -} \\{{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} -} \\{{\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}}}\end{pmatrix}}} \\{{^{{\alpha}\; t} +}} \\{{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} +} \\{{\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}} +} \\{{\cos \; \Phi_{+ 1}} + {\sin \; \Phi_{+ 1}} +} \\{{\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}}}\end{pmatrix}}} \\{{^{{- {\alpha}}\; t} +}} \\{{\frac{}{4}\begin{pmatrix}{{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} -} \\{{\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}} +} \\{{\cos \; \Phi_{+ 1}} + {\sin \; \Phi_{+ 1}} -} \\{{\cos \; \Phi_{- 1}} - {\sin \; \Phi_{- 1}}}\end{pmatrix}}} \\{{^{{\alpha}\; t} +}} \\{{\frac{}{4}\begin{pmatrix}{{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} -} \\{{\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}} -} \\{{\cos \; \Phi_{+ 1}} - {\sin \; \Phi_{+ 1}} +} \\{{\cos \; \Phi_{- 1}} + {\sin \; \Phi_{- 1}}}\end{pmatrix}}} \\{^{{- {\alpha}}\; t}} \\{= {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}}} \right)^{{- {\alpha}}\; t}} -}} \\{{{\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}}} \right)^{\; {\pi/2}}^{{- {\alpha}}\; t}} -}} \\{{{\frac{1}{2}\left( {{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}}} \right)^{{\alpha}\; t}} +}} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}}} \right)^{\; {\pi/2}}^{{\alpha}\; t}}}\end{matrix} & (141)\end{matrix}$

Combining S_(−1,−1)(t) of Eq. 14 and S*_(+1,−1)(t) of Eq. 141 oneobtains

$\begin{matrix}{{T_{{+ 1},{- 1}}\left( {t,M} \right)} \propto {\underset{j = 0}{\overset{K}{\otimes}}{\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},j}} + {\cos \; \Phi_{{- 1},j}}} \right)^{\; M_{j}\alpha_{j}t}} +} \\{{\frac{M_{j}}{2}\left( {{\sin \; \Phi_{{+ 1},j}} - {\sin \; \Phi_{{- 1},j}}} \right)^{\; {\pi/2}}^{\; M_{j}\alpha_{j}t}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},j}} + {\sin \; \Phi_{{- 1},j}}} \right)^{{- }\; M_{j}\alpha_{j}t}} -} \\{\frac{M_{j}}{2}\left( {{\cos \; \Phi_{{+ 1},j}} - {\cos \; \Phi_{{- 1},j}}} \right)^{\; {\pi/2}}^{\; M_{j}\alpha_{j}t}}\end{pmatrix}.}}} & (142)\end{matrix}$

With Eq. 142, the complex time domain signal for the edited sub spectrumrepresented by M=[1 1], is proportional to

$\begin{matrix}{{{T_{{+ 1},{- 1}}\left( {t,M} \right)} \propto {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},1}} + {\cos \; \Phi_{{- 1},1}}} \right)^{\; \alpha_{1}t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},1}} - {\sin \; \Phi_{{- 1},1}}} \right)^{\; {\pi/2}}^{\; \alpha_{1}t}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},1}} + {\sin \; \Phi_{{- 1},1}}} \right)^{{- }\; \alpha_{1}t}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},1}} - {\cos \; \Phi_{{- 1},1}}} \right)^{\; {\pi/2}}^{{- }\; \alpha_{1}\; t}}\end{pmatrix} \otimes \begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}}} \right)^{\; {\alpha \;}_{0}t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}}} \right)^{\; {\pi/2}}^{\; \alpha_{0}t}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}}} \right)^{{- }\; \alpha_{0}t}} -} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}}} \right)^{\; {\pi/2}}^{{- }\; \alpha_{0}t}}\end{pmatrix}}},} & (143)\end{matrix}$

which is equivalent to

$\begin{matrix}{{T_{{+ 1},{- 1}}\left( {t,M} \right)} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{{{({\alpha_{1} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}\end{pmatrix}^{\; {({\alpha_{1} - \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} - \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} - \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} - \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}{^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}.}}}} & (144)\end{matrix}$

FT along the GFT dimension reveals mixed phase peaks at (i) the desiredposition (α₁+α₀), (ii) the quad position −(α₁+α₀) of the desired peak,(iii) the ‘cross-talk’ position (−α₁+α₀) and the quad position (α₁−α₀)of the cross-talk peak. The ‘cross talk’ between GFT NMR sub-spectraresults in peaks that are located at linear combinations of chemicalshifts other than the desired one. Under the condition of identicalsecondary phase shifts for forward and backward samplings (Eq. 35), Eq.144 simplifies to

T_(+1,−1)(t,M)∝(cos Φ_(1,1) cos Φ_(1,0))e^(i(α) ¹ ^(+α) ⁰ ^()t)−(cosΦ_(1,1) sin Φ_(1,0))e^(i(α) ¹ ^(−α) ⁰ ^()t)−(sin Φ_(1,1) cosΦ_(1,0))e^(i(−α) ^(i) ^(+α) ⁰ ^()t−(sin Φ) _(1,1) cos Φ_(1,0))e^(i(−α) ¹^(+α) ⁰ ^()t)+(sin Φ_(1,1) sin Φ_(1,0))e^(−i(α) ¹ ^(+α) ⁰ ^()t)  (145).

FT along the GFT dimension reveals absorptive peaks at (i) the desiredposition −(α₁+α₀), (ii) the quad position −(α₁+α₀) of the desired peak,(iii) the cross-talk position (−α₁+α₀) and the quad position (α₁−α₀) ofthe cross-talk peak. Under the condition of secondary phase shiftsindependent of n (Eq. 54), the peak components of Eq. 144 does notchange. Under the condition of identical secondary phase shifts (Eq.73), Eq. 144 simplifies to

T_(+1,−1)(t,M)∝cos Φ₁ cos Φ₀e^(i(α) ¹ ^(+α) ⁰ ^()t)−(cos Φ₁ sinΦ₀)e^(i(α) ¹ ^(−α) ⁰ ^()t)−(sin Φ₁ cos Φ₀)e^(i(−α) ¹ ^(+α) ⁰ ^()t)+(sinΦ₁ cos Φ₀)e^(i(−α) ¹ ^(+α) ⁰ ^()t)+(sin Φ₁ sin Φ₀)e^(−i(α) ¹ ^(+α) ⁰^()t)  (146).

(c₊₃,c⁻³)-Sampling (PMS)

With S_(+3,−3)(t) of Eq. 18 the complex conjugate is proportional to

$\begin{matrix}{{{S_{{+ 3},{- 3}}^{*}(t)} \propto {\begin{bmatrix}1 & {- }\end{bmatrix}D_{{+ 3},{- 3}}{C_{{+ 3},{- 3}}(t)}}} = {{{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & {- }\end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 3}(t)} \\{c_{- 3}(t)}\end{bmatrix}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}{{- 1} + } & {{- 1} - }\end{bmatrix}}\begin{bmatrix}{- {\sin \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 3}} \right)}} \\{- {\sin \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 3}} \right)}}\end{bmatrix}} = {{{\frac{1}{\sqrt{2}}\left( {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) - {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right)\cos \left( {\alpha \; t} \right)} + {\frac{1}{\sqrt{2}}\left( {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) - {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right){\sin \left( {\alpha \; t} \right)}}} = {{{\frac{1}{\sqrt{2}}\left( {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) - {\left( {{\sin \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\sin \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right)\frac{^{\; \alpha \; t} + ^{{- }\; \alpha \; t}}{2}} + {\frac{1}{\sqrt{2}}\left( {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} - {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right) - {\left( {{\cos \left( {\frac{\pi}{4} + \Phi_{+ 3}} \right)} + {\cos \left( {\frac{\pi}{4} + \Phi_{- 3}} \right)}} \right)}} \right)\frac{^{\; \alpha \; t} - ^{{- }\; \alpha \; t}}{2}}} = {{{\frac{1}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} + {\cos \mspace{11mu} \Phi_{- 3}} + {\sin \; \Phi_{- 3}} + {\cos \; \Phi_{+ 3}} - {\sin \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}} - {\sin \; \Phi_{- 3}}} \right)^{{- }\; \alpha \; t}} + {\frac{1}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} + {\cos \mspace{11mu} \Phi_{- 3}} + {\sin \; \Phi_{- 3}} - {\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}} + {\sin \; \Phi_{- 3}}} \right)^{\; \alpha \; t}} + {\frac{\;}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} - {\cos \mspace{11mu} \Phi_{- 3}} - {\sin \; \Phi_{- 3}} - {\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}} - {\sin \; \Phi_{- 3}}} \right)^{{- }\; \alpha \; t}} + {\frac{\;}{4}\left( {{\cos \; \Phi_{+ 3}} + {\sin \; \Phi_{+ 3}} - {\cos \mspace{11mu} \Phi_{- 3}} - {\sin \; \Phi_{- 3}} + {\cos \; \Phi_{+ 3}} - {\sin \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}} + {\sin \; \Phi_{- 3}}} \right)^{\; \alpha \; t}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}} \right)^{{- }\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}} \right)^{\; {\pi/2}}^{{- }\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 3}} + {\sin \; \Phi_{- 3}}} \right)^{\; \alpha \; t}} - {\frac{1}{2}\left( {{\cos \; \Phi_{+ 3}} - {\cos \; \Phi_{- 3}}} \right)^{\; {\pi/2}}{^{\; \alpha \; t}.}}}}}}}}} & (147)\end{matrix}$

Combining S_(+3,−3)(t) of Eq. 18 and S*₊ _(3,−3)(t) of Eq. 147 oneobtains

$\begin{matrix}{{T_{{+ 3},{- 3}}\left( {t,M} \right)} \propto {\underset{j = 0}{\overset{K}{\otimes}}{\begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},j}} + {\cos \; \Phi_{{- 1},j}}} \right)^{\; M_{j}\alpha_{j}t}} +} \\{{\frac{M_{j}}{2}\left( {{\sin \; \Phi_{{+ 1},j}} - {\sin \; \Phi_{{- 1},j}}} \right)^{\; {\pi/2}}^{\; M_{j}\; \alpha_{j}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},j}} + {\sin \; \Phi_{{- 1},j}}} \right)^{{- }\; M_{j}\alpha_{j}t}} +}\end{matrix} \\{\frac{M_{j}}{2}\left( {{\cos \; \Phi_{{+ 1},j}} - {\cos \; \Phi_{{- 1},j}}} \right)^{\; {\pi/2}}^{{- }\; M_{j}\alpha_{j}t}}\end{pmatrix}.}}} & (148)\end{matrix}$

With Eq. 148, the complex time domain signal for the edited sub spectrumrepresented by M=[1 1], is proportional to

$\begin{matrix}{{{T_{{+ 3},{- 3}}\left( {t,M} \right)} \propto {\begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},1}} + {\cos \; \Phi_{{- 1},1}}} \right)^{\; \alpha_{1}\; t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},1}} - {\sin \; \Phi_{{- 1},1}}} \right)^{\; {\pi/2}}^{\; \alpha_{1}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},1}} + {\sin \; \Phi_{{- 1},1}}} \right)^{{- }\; \alpha_{1}t}} +}\end{matrix} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},1}} - {\cos \; \Phi_{{- 1},1}}} \right)^{\; {\pi/2}}^{{- }\; \alpha_{1}t}}\end{pmatrix} \otimes \begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}}} \right)^{\; \alpha_{0}\; t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}}} \right)^{\; {\pi/2}}^{\; \alpha_{0}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}}} \right)^{{- }\; \alpha_{0}t}} +}\end{matrix} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}}} \right)^{\; {\pi/2}}^{{- }\; \alpha_{0}t}}\end{pmatrix}}},} & (149)\end{matrix}$

which is equivalent to

$\begin{matrix}{{T_{{+ 3},{- 3}}\left( {t,M} \right)} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} - \alpha_{0}})}}\; t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- {{({\alpha_{1} + \alpha_{0}})}}}\; t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}{^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}.}}}} & (150)\end{matrix}$

FT along the GFT dimension reveals the same peak components asT_(+1,−1)(t,M). Under the condition of identical secondary phase shiftsfor forward and backward samplings (Eq. 35), Eq.150 simplifies to

T_(+3,−3)(t,M)∝(cos Φ_(3,1) cos Φ_(3,0))e^(i(α) ¹ ^(+α) ⁰ ^()t)+(cosΦ_(3,1) sin Φ_(3,0))e^(i(α) ¹ ^(−α) ⁰ ^()t)+(sin Φ_(3,1) cosΦ_(3,0))e^(i(−α) ^(i) ^(+α) ⁰ ^()t)+(sin Φ_(3,1) cos Φ_(3,0))e^(i(−α) ¹^(+α) ⁰ ^()t)  (151).

and under the condition of identical secondary phase shifts (Eq. 35),Eq. 150 simplifies to

T_(+3,−3)(t,M)∝cos Φ₁ cos Φ₀)e^(i(α) ¹ ^(+α) ⁰ ^()t)+(cos Φ₁ sinΦ₀)e^(i(α) ¹ ^(−α) ⁰ ^()t)+(sin Φ₁ cos Φ₀)e^(i(−α) ^(i) ^(+α) ⁰^()t)+(sin Φ₁ cos Φ₀)e^(i(−α) ¹ ^(+α) ⁰ ^()t)  (152).

(c₊₁,c⁻¹,c₊₃,c⁻³)-sampling (DPMS)

Starting with Eq. 21, the complex signal for two indirect dimensions isproportional to

S _(+1,−1,+3,−3)(t ₁ ,t ₀)=└S _(+1,−1)(t ₁)+S ₊3,−3(t ₁)┘

└S _(+1,−1)(t ₀)+S _(+3,−3)(t ₀)┘∝S ₊ _(1,−1)(t ₁)S _(+1,−1)(t ₀)+S_(+1, −1)(t ₁)S _(+3,−3)(t ₀)+S _(+3,−3)(t ₁)S _(+1, −1)(t ₀)+S_(+3,−3)(t ₁)S _(+3,−3)(t ₀)  (153).

DPMS for two jointly sampled chemical shifts requires all combinationsof π/4 and π/4-shifted mirrored sampling of the two chemical shifts. Thefirst and the last term in Eq. 153 are represented, respectively, byEqs. 144 and 150. It is then straightforward to show that the signals ofthe spectrum corresponding to the second term in Eq. 153 where α₀ issampled as (C₊₃,c⁻³) and α₁ as (c₊₁,c⁻¹), is proportional to

$\begin{matrix}{{{T_{{+ 1},{- 1}}\left( {t,M} \right)} \otimes {T_{{+ 3},{- 3}}\left( {t,M} \right)}} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} - \alpha_{0}})}}\; t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} - \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- {{({\alpha_{1} + \alpha_{0}})}}}\; t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}{^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}.}}}} & (154)\end{matrix}$

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq.154 simplifies to

T_(+1,−1)(t,M)

T_(+1,−3)(t,M)∝(cos Φ_(1,1) cos Φ_(3,0))e^(i(α) ¹ ^(+α) ⁰ ^()t)+(cosΦ_(1,1) sin Φ_(3,0))e^(i(α) ¹ ^(−α) ⁰ ^()t)−(sin Φ_(1,1) cosΦ_(3,0))e^(i(−α) ^(i) ^(+α) ⁰ ^()t)−(sin Φ_(1,1) cos Φ_(3,0))e^(i(−α) ¹^(+α) ⁰ ^()t)  (155).

Under the condition of secondary phase shifts independent of n (Eq. 54),the peak components of Eq. 154 remain unchanged. Under the condition ofidentical secondary phase shifts (Eq. 73), Eq. 154 simplifies to

T_(+1,−1)(t,M)

T_(+3,−3)(t,M)∝cos ² Φe^(i(α) ¹ ^(+α) ⁰ ^()t)+(cos Φ_(1,1) sinΦ_(3,0))e^(i(α) ¹ ^(−α) ⁰ ^()t)−(sin Φcos Φ)e^(i(−α) ^(i) ^(+α) ⁰^()t)−(sin ² Φe^(i(−α) ¹ ^(+α) ⁰ ^()t)  (156).

The signals of the spectrum corresponding to the third term in Eq. 153is proportional to

$\begin{matrix}{{{T_{{+ 3},{- 3}}\left( {t,M} \right)} \otimes {T_{{+ 1},{- 1}}\left( {t,M} \right)}} \propto {{\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {({\alpha_{1} - \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} - \alpha_{0}})}}\; t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} - \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} - {\frac{1}{4}\begin{pmatrix}{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- {{({\alpha_{1} + \alpha_{0}})}}}\; t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({{- \alpha_{1}} + \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}} - {\frac{1}{4}\begin{pmatrix}{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}}}\end{pmatrix}^{\; \pi}{^{{- }\; {({\alpha_{1} + \alpha_{0}})}t}.}}}} & (157)\end{matrix}$

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq.157 simplifies to

T_(+3,−3)(t,M)

T_(+1,−1)(t,M)∝(cos Φ_(3,1) cos Φ_(1,0))e^(i(α) ¹ ^(+α) ⁰ ^()t)−(cosΦ_(3,1) sin Φ_(1,0))e^(i(α) ¹ ^(−α) ⁰ ^()t)+(sin Φ_(3,1) cosΦ_(1,0))e^(i(−α) ^(i) ^(+α) ⁰ ^()t)−(sin Φ_(3,1) cos Φ_(1,0))e^(i(−α) ¹^(+α) ⁰ ^()t)  (158).

Under the condition of secondary phase shifts independent of n (Eq. 54),the peak components of Eq. 157 remain unchanged. Under the condition ofidentical secondary phase shifts (Eq. 73), Eq. 157 simplifies to

T_(+3,−3)(t,M)

T_(+1,−1)(t,M)∝cos Φ₁ cos Φ₀)e^(i(α) ¹ ^(+α) ⁰ ^()t)−(cos Φ_(3,1) sinΦ_(1,0))e^(i(α) ¹ ^(−α) ⁰ ^()t)+(sin Φ₁ cos Φ₀)e^(i(−α) ^(i) ^(+α) ⁰^()t)−(sin Φ₁ cos Φ₀)e^(i(−α) ¹ ^(+α) ⁰ ^()t)  (159).

Addition of Eqs. 144, 150, 154 and 157 gives the complex signal for(c₊₁,c⁻¹,c₊₃,c⁻³)-sampling for two jointly sampled chemical shifts andis proportional to

$\begin{matrix}{{T_{{+ 1},{- 1},{+ 3},{- 3}}\left( {t,M} \right)} \propto {{\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{{{({\alpha_{1} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{{{({\alpha_{1} - \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{\; {({\alpha_{1} - \alpha_{0}})}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{{{({\alpha_{1} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({\alpha_{1} - \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{{{({\alpha_{1} - \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; s\; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} - {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 1},1}\sin \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 1},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 3},1}\sin \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},1}\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; {\pi/2}}^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}} + {\frac{1}{4}\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 1},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{+ 1},1}\cos \; \Phi_{{- 3},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 1},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 1},0}} +}\end{matrix} \\{{\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{+ 3},1}\cos \; \Phi_{{- 3},0}} -}\end{matrix} \\{{\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},1}\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; \pi}{^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}.}}}} & (160)\end{matrix}$

Under the condition of identical secondary phase shifts for forward andbackward samplings (Eq. 35), Eq.160 simplifies to

$\begin{matrix}{{T_{{+ 1},{- 1},{+ 3},{- 3}}\left( {t,M} \right)} \propto {{\left( {{\cos \; \Phi_{1,1}\cos \; \Phi_{1,0}} + {\cos \; \Phi_{1,1}\cos \; \Phi_{3,0}} + {\cos \; \Phi_{3,1}\cos \; \Phi_{1,0}} + {\cos \; \Phi_{3,1}\cos \; \Phi_{3,0}}} \right)^{{{({\alpha_{1} + \alpha_{0}})}}t}} - {\left( {{\cos \; \Phi_{1,1}\sin \; \Phi_{1,0}} - {\cos \; \Phi_{1,1}\sin \; \Phi_{3,0}} + {\cos \; \Phi_{3,1}\sin \; \Phi_{1,0}} - {\cos \; \Phi_{3,1}\sin \; \Phi_{3,0}}} \right)^{{{({\alpha_{1} - \alpha_{0}})}}t}} - {\left( {{\sin \; \Phi_{1,1}\cos \; \Phi_{1,0}} + {\sin \; \Phi_{1,1}\cos \; \Phi_{3,0}} - {\sin \; \Phi_{3,1}\cos \; \Phi_{1,0}} - {\sin \; \Phi_{3,1}\cos \; \Phi_{3,0}}} \right)^{{{({{- \alpha_{1}} + \alpha_{0}})}}t}} + {\left( {{\sin \; \Phi_{1,1}\sin \; \Phi_{1,0}} - {\sin \; \Phi_{1,1}\sin \; \Phi_{3,0}} - {\sin \; \Phi_{3,1}\sin \; \Phi_{1,0}} + {\sin \; \Phi_{3,1}\sin \; \Phi_{3,0}}} \right){^{{- {{({\alpha_{1} + \alpha_{0}})}}}t}.}}}} & (161)\end{matrix}$

Two dimensional FT reveals absorptive peaks at the desired, cross-talkand the quad positions. Under the condition of secondary phase shiftsindependent of n (Eq. 54), Eq.160 simplifies to

$\begin{matrix}{{{F_{C}\left( {S_{{+ 1},{- 1},{+ 3},{- 3}}\left( {t_{1},t_{0}} \right)} \right)} \propto {{\left( {{\cos \; \Phi_{+ {,1}}\cos \; \Phi_{+ {,0}}} + {\cos \; \Phi_{+ {,1}}\cos \; \Phi_{- {,0}}} + {\cos \; \Phi_{- {,1}}\cos \; \Phi_{+ {,0}}} + {\cos \; \Phi_{- {,1}}\cos \; \Phi_{- {,0}}}} \right)^{{{({\alpha_{1} + \alpha_{0}})}}t}} + {\left( {{\cos \; \Phi_{+ {,1}}\sin \; \Phi_{+ {,0}}} - {\cos \; \Phi_{+ {,1}}\sin \; \Phi_{- {,0}}} + {\cos \; \Phi_{- {,1}}\sin \; \Phi_{+ {,0}}} - {\cos \; \Phi_{- {,1}}\sin \; \Phi_{- {,0}}}} \right)^{\; {\pi/2}}^{{{({\alpha_{1} + \alpha_{0}})}}t}} + {\left( {{\sin \; \Phi_{+ {,1}}\cos \; \Phi_{+ {,0}}} + {\sin \; \Phi_{+ {,1}}\cos \; \Phi_{- {,0}}} - {\sin \; \Phi_{- {,1}}\cos \; \Phi_{+ {,0}}} - {\sin \; \Phi_{- {,1}}\cos \; \Phi_{- {,0}}}} \right)^{\; {\pi/2}}^{{{({\alpha_{1} + \alpha_{0}})}}t}} + {\left( {{\sin \; \Phi_{+ {,1}}\sin \; \Phi_{+ {,0}}} - {\sin \; \Phi_{+ {,1}}\sin \; \Phi_{- {,0}}} - {\sin \; \Phi_{- {,1}}\sin \; \Phi_{+ {,0}}} + {\sin \; \Phi_{- {,1}}\sin \; \Phi_{- {,0}}}} \right)^{\; \pi}^{{{({\alpha_{1} + \alpha_{0}})}}t}}}},} & (162)\end{matrix}$

Two dimensional FT reveals a single mixed phase peak at the desiredposition. This proves that, imbalance between forward and backwardsamplings can not eliminate dispersive components. Under the conditionof identical secondary phase shifts (Eq. 73), Eq. 160 simplifies to

F_(C)(S_(+1, −1,+3,−3)(t₁,t₀))∝4 cos Φ₁ cos Φ₀e^(i(α) ¹ ^(+α) ⁰^()t)  (163).

Two dimensional FT reveals a single peak, 4 cos Φ₁ cos Φ₀e^(i(α) ¹ ^(+α)⁰ ^()t), which is purely absorptive and located at the desired linearcombination of chemical shift (α₁+α₀).

Arbitrary Subset of Jointly Sampled Chemical Shifts

As in multi-dimensional NMR, multiple PMS in GFT results for the complextime domain signal in a sum over products of all possible permutationsof cos φ_(±n,j) and sin φ_(±n,j).

(c₊₀,c₊₂)-Sampling

Combining S_(+0,+2)(t) of Eq. 3 and S*_(+0,+2) (t) of Eq. 136 oneobtains

$\begin{matrix}{{{T_{{+ 0},{+ 2}}\left( {t,M} \right)} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{+ 2},K}}} \right)^{\; M_{K}\; \alpha_{K}\; t}} +} \\{{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{+ 2},K}}} \right)^{{- }\; M_{K}\alpha_{K}t}} -} \\{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{pmatrix} \otimes \begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)^{\; M_{K - 1}\; \alpha_{K - 1}\; t}} +} \\{{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)^{{- }\; M_{K - 1}\alpha_{K - 1}t}} -}\end{matrix} \\{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{pmatrix} \otimes \ldots \otimes \begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{+ 2},0}}} \right)^{\; M_{0}\; \alpha_{0}\; t}} +} \\{{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{+ 2},0}}} \right)^{{- }\; M_{0}\alpha_{0}t}} -}\end{matrix} \\{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{pmatrix}}},} & (164)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c⁻⁰,c⁻²)-Sampling

With S_(−0,−2)(t) of Eq. 7, the complex conjugate is proportional to

$\begin{matrix}{{{S_{{- 0},{- 2}}^{*}(t)} \propto {Q^{*}D_{{- 0},{- 2}}{C_{{- 0},{- 2}}(t)}}} = {{\begin{bmatrix}1 & {- }\end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}{\quad{\begin{bmatrix}{c_{- 0}(t)} \\{c_{- 2}(t)}\end{bmatrix} = {{\begin{bmatrix}1 & {- }\end{bmatrix}\begin{bmatrix}c_{- 0} \\c_{- 2}\end{bmatrix}} = {{\begin{bmatrix}1 & {- }\end{bmatrix}\begin{bmatrix}{\cos \left( {{\alpha \; t} - \Phi_{- 0}} \right)} \\{\sin \left( {{\alpha \; t} - \Phi_{- 2}} \right)}\end{bmatrix}} = {{\left( {{\cos \; \Phi_{- 0}{\cos \left( {\alpha \; t} \right)}} + {\sin \; \Phi_{- 0}{\sin \left( {\alpha \; t} \right)}}} \right) + {\left( {{\sin \; \Phi_{- 2}{\cos \left( {\alpha \; t} \right)}} - {\cos \; \Phi_{- 2}{\sin \left( {\alpha \; t} \right)}}} \right)}} = {{{\left( {{\cos \; \Phi_{- 0}} + {\; \sin \; \Phi_{- 2}}} \right){\cos \left( {\alpha \; t} \right)}} + {\left( {{\sin \; \Phi_{- 0}} - {\; \cos \; \Phi_{- 2}}} \right)\sin \; \left( {\alpha \; t} \right)}} = {{{\left( {{\cos \; \Phi_{- 0}} + {\; \sin \; \Phi_{- 2}}} \right)\frac{^{\; \alpha \; t} + ^{{- }\; \alpha \; t}}{2}} + {\left( {{\sin \; \Phi_{- 0}} - {\; \cos \; \Phi_{- 2}}} \right)\frac{^{\; \alpha \; t} - ^{{- }\; \alpha \; t}}{2\; }}} = {{{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} - {\frac{}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} + {\frac{\;}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}}} = {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}{^{{- }\; \alpha \; t}.}}}}}}}}}}}}} & (165)\end{matrix}$

Combining S_(−0,−2)(t) of Eq. 7 and S*_(−0,−2) (t) of Eq. 165 oneobtains

$\begin{matrix}{{T_{{- 0},{- 2}}\left( {t,M} \right)} = {\begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)^{\; M_{K}\alpha_{K}t}} -} \\{{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)^{{- }\; M_{K}\alpha_{K}t}} +}\end{matrix} \\{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{pmatrix} \otimes \begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}} \right)^{\; M_{K - 1}\alpha_{K - 1}t}} -} \\{{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}} \right)^{{- }\; M_{K - 1}\alpha_{K - 1}t}} +}\end{matrix} \\{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}\begin{matrix}\begin{matrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)^{\; M_{0}\alpha_{0}\; t}} -} \\{{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}\; t}} +}\end{matrix} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{- 2},0}}} \right)^{{- }\; M_{0}\alpha_{0}\; t}} +}\end{matrix} \\{\frac{M_{0}\;}{2}\left( {{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{pmatrix}.}}} & (166)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c₊₀,c₊₂,c⁻⁰,c⁻²)-Sampling

As for Eq. 10, addition of S*_(+0, +2) (t) and S*_(−0,−2) (t) yields

$\begin{matrix}{{S_{{+ 0},{+ 2},{- 0},{- 2}}^{*}(t)} = {{{S_{{+ 0},{+ 2}}^{*}(t)} + {S_{{- 0},{- 2}}^{*}(t)}} \propto {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} - {\cos \; \Phi_{- 2}}} \right)^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} + {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}^{\; \alpha \; t}} + {\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{+ 2}} + {\cos \; \Phi_{- 0}} + {\cos \; \Phi_{- 2}}} \right)^{{- }\; \alpha \; t}} - {\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{+ 2}} - {\sin \; \Phi_{- 0}} - {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}{^{{- }\; \alpha \; t}.}}}}} & (167)\end{matrix}$

Combining S_(+0,+2,−0,−2)(t) of Eq. 10 and S*₊ _(0, +2,−0,−2)(t) of Eq.167 one obtains

$\begin{matrix}{{T_{{+ 0},{+ 2},{- 0},{- 2}}(t)} \propto {{\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{+ 2},K}} +} \\{{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{- 2},K}}}\end{pmatrix}^{\; M_{K}\alpha_{K}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{+ 2},K}} -} \\{{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{- 2},K}}}\end{pmatrix}^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{+ 2},K}} +} \\{{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{- 2},K}}}\end{pmatrix}^{{- }\; M_{K}\alpha_{K}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{+ 2},K}} -} \\{{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{- 2},K}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}} \\{^{\; M_{K - 1}\alpha_{K - 1}t} +} \\\begin{matrix}\begin{matrix}{\frac{M_{K - 1}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{- 2},K}}}\end{pmatrix}} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} -}\end{matrix} \\{\frac{M_{K - 1}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix}} {\quad {{\otimes {\ldots \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{+ 2},0}} +} \\{{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{- 2},0}}}\end{pmatrix}^{\; M_{0}\alpha_{0}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{0}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{+ 2},0}} -} \\{{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{- 2},0}}}\end{pmatrix}^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{+ 2},0}} +} \\{{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{- 2},0}}}\end{pmatrix}^{{- }\; M_{0}\alpha_{0}t}} -}\end{matrix} \\{\frac{M_{0}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{+ 2},0}} -} \\{{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{- 2},0}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}}},}}}} & (168)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c₊₁,c⁻¹)-Sampling (PMS)

Combining S_(+1,−1)(t) of Eq. 14 and S*_(+1,−1)(t) of Eq. 141 oneobtains

$\begin{matrix}{{T_{{+ 1},{- 1}}\left( {t,M} \right)} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},K}} + {\cos \; \Phi_{{- 1},K}}} \right)^{\; M_{K}\alpha_{K}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{+ 1},K}} - {\sin \; \Phi_{{- 1},K}}} \right)^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},K}} + {\sin \; \Phi_{{- 1},K}}} \right)^{{- }\; M_{K}\alpha_{K}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\left( {{\cos \; \Phi_{{+ 1},K}} - {\cos \; \Phi_{{- 1},K}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},{K - 1}}} + {\cos \; \Phi_{{- 1},{K - 1}}}} \right)} \\{^{\; M_{K - 1}\alpha_{K - 1}t} +} \\\begin{matrix}\begin{matrix}{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{+ 1},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}}} \right)} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} -} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}}} \right)} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} -}\end{matrix} \\{\frac{M_{K - 1}}{2}\left( {{\cos \; \Phi_{{+ 1},{K - 1}}} - {\cos \; \Phi_{{- 1},{K - 1}}}} \right)} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}}} \right)^{\; M_{0}\alpha_{0}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}}} \right)^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} -} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}}} \right)^{{- }\; M_{0}\alpha_{0}t}} -}\end{matrix} \\{\frac{M_{0}}{2}\left( {{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}.}}} & (169)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c₊₃,c⁻³)-Sampling (PMS)

Combining S_(+3,−3)(t) of Eq. 18 and S*_(+3,−3)(t) of Eq. 147 oneobtains

$\begin{matrix}{{T_{{+ 3},{- 3}}\left( {t,M} \right)} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}} \right)^{\; M_{K}\alpha_{K}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}} \right)^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},K}} + {\sin \; \Phi_{{- 3},K}}} \right)^{{- }\; M_{K}\alpha_{K}t}} +}\end{matrix} \\{\frac{M_{K}}{2}\left( {{\cos \; \Phi_{{+ 3},K}} - {\cos \; \Phi_{{- 3},K}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}} \right)} \\{^{\; M_{K - 1}\alpha_{K - 1}t} +} \\\begin{matrix}\begin{matrix}{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}}} \right)} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +} \\{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}}} \right)} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} +}\end{matrix} \\{\frac{M_{K - 1}}{2}\left( {{\cos \; \Phi_{{+ 3},{K - 1}}} - {\cos \; \Phi_{{- 1},{K - 1}}}} \right)} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}} \right)^{\; M_{0}\alpha_{0}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}} \right)^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} +} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{{+ 3},0}} + {\sin \; \Phi_{{- 3},0}}} \right)^{{- }\; M_{0}\alpha_{0}t}} +}\end{matrix} \\{\frac{M_{0}}{2}\left( {{\cos \; \Phi_{{+ 3},0}} - {\cos \; \Phi_{{- 3},0}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}.}}} & (170)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c₊₁,c⁻¹,c₊₃,C⁻³)-Sampling (DPMS)

As for Eq. 21, addition of S*_(+1,−3) (t) and S*_(+3,−3) (t) yields

$\begin{matrix}{{S_{{+ 1},{- 1},{+ 3},{- 3}}^{*}(t)} = {{{S_{{+ 1},{- 1}}^{*}(t)} + {S_{{+ 3},{- 3}}^{*}(t)}} \propto {{{- \frac{1}{2}}\begin{pmatrix}{{\sin \; \Phi_{+ 1}} + {\sin \; \Phi_{- 1}} -} \\{{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}\end{pmatrix}^{{\alpha}\; t}} + {\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{+ 1}} - {\cos \; \Phi_{- 1}} -} \\{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}\end{pmatrix}^{\frac{\pi}{2}}^{{\alpha}\; t}} + {\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{+ 1}} + {\cos \; \Phi_{- 1}} +} \\{{\cos \; \Phi_{+ 3}} + {\cos \; \Phi_{- 3}}}\end{pmatrix}^{{- {\alpha}}\; t}} - {\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{+ 1}} - {\sin \; \Phi_{- 1}} +} \\{{\sin \; \Phi_{+ 3}} - {\sin \; \Phi_{- 3}}}\end{pmatrix}^{\frac{\pi}{2}}{^{{- {\alpha}}\; t}.}}}}} & (171)\end{matrix}$

Combining S+_(1,−1,+3,−3)(t) of Eq. 21 and S*_(−1, −1, +3,−3)(t) of Eq.171 one obtains

$\begin{matrix}{{{T_{{+ 1},{- 1},{+ 3},{- 3}}(t)} \propto {\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},K}} + {\cos \; \Phi_{{- 1},K}} +} \\{{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}}\end{pmatrix}^{\; M_{K}\alpha_{K}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},K}} - {\sin \; \Phi_{{- 1},K}} +} \\{{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}}\end{pmatrix}^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} -} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},K}} + {\sin \; \Phi_{{- 1},K}} -} \\{{\sin \; \Phi_{{+ 3},K}} - {\sin \; \Phi_{{- 3},K}}}\end{pmatrix}^{{- }\; M_{K}\alpha_{K}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},K}} - {\cos \; \Phi_{{- 1},K}} -} \\{{\cos \; \Phi_{{+ 3},K}} + {\cos \; \Phi_{{- 3},K}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},{K - 1}}} + {\cos \; \Phi_{{- 1},{K - 1}}} +} \\{{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}} \\{^{\; M_{K - 1}\alpha_{K - 1}t} +} \\\begin{matrix}\begin{matrix}{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},{K - 1}}} - {\sin \; \Phi_{{- 1},{K - 1}}} +} \\{{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} -} \\{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},{K - 1}}} + {\sin \; \Phi_{{- 1},{K - 1}}} -} \\{{\sin \; \Phi_{{+ 3},{K - 1}}} - {\sin \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},{K - 1}}} - {\cos \; \Phi_{{+ 1},{K - 1}}} -} \\{{\cos \; \Phi_{{+ 3},{K - 1}}} + {\cos \; \Phi_{{- 3},{K - 1}}}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix} \otimes \ldots \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},0}} + {\cos \; \Phi_{{- 1},0}} +} \\{{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\; M_{0}\alpha_{0}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},0}} - {\sin \; \Phi_{{- 1},0}} +} \\{{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}} -} \\{{\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 1},0}} + {\sin \; \Phi_{{- 1},0}} -} \\{{\sin \; \Phi_{{+ 3},0}} - {\sin \; \Phi_{{- 3},0}}}\end{pmatrix}^{{- }\; M_{0}\alpha_{0}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 1},0}} - {\cos \; \Phi_{{- 1},0}} -} \\{{\cos \; \Phi_{{+ 3},0}} + {\cos \; \Phi_{{- 3},0}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}}},} & (172)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c₊₀,c⁻²)-Sampling (PMS)

With S_(+0,−2)(t) of Eq. 25, the complex conjugate is proportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 0} - 2}^{*}(t)} \propto {Q^{*}D_{{+ 0},{- 2}}{C_{{+ 0},{- 2}}(t)}}} = {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 0}(t)} \\{c_{- 2}(t)}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}c_{+ 0} \\c_{- 2}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}{\cos \left( {{\alpha \; t} + \Phi_{+ 0}} \right)} \\{\sin \left( {{\alpha \; t} - \Phi_{- 2}} \right)}\end{bmatrix}}} \\{= {\begin{pmatrix}{{\cos \; \Phi_{+ 0}{\cos \left( {\alpha \; t} \right)}} -} \\{\sin \; \Phi_{+ 0}{\sin \left( {\alpha \; t} \right)}}\end{pmatrix} +}} \\{{\begin{pmatrix}{{\sin \; \Phi_{- 2}{\cos \left( {\alpha \; t} \right)}} -} \\{\cos \; \Phi_{- 2}{\sin \left( {\alpha \; t} \right)}}\end{pmatrix}}} \\{= {{\left( {{\cos \; \Phi_{+ 0}} + {sin\Phi}_{- 2}} \right){\cos \left( {\alpha \; t} \right)}} -}} \\{{\left( {{\sin \; \Phi_{+ 0}} + {cos\Phi}_{- 2}} \right){\sin \left( {\alpha \; t} \right)}}} \\{= {{\left( {{\cos \; \Phi_{+ 0}} + {sin\Phi}_{- 2}} \right)\frac{^{{\alpha}\; t} + ^{{- {\alpha}}\; t}}{2}} -}} \\{{\left( {{\sin \; \Phi_{+ 0}} + {cos\Phi}_{- 2}} \right)\frac{^{{\alpha}\; t} + ^{{- {\alpha}}\; t}}{2}}} \\{= {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}}} \right)^{{\alpha}\; t}} +}} \\{{{\frac{}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}}} \right)^{{\alpha}\; t}} +}} \\{{{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}} \right)^{{- {\alpha}}\; t}} -}} \\{{\frac{}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}} \right)^{{- {\alpha}}\; t}}} \\{= {{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}}} \right)^{{\alpha}\; t}} +}} \\{{{\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}^{{\alpha}\; t}} +}} \\{{{\frac{1}{2}\left( {{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}}} \right)^{{- {\alpha}}\; t}} -}} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}}} \right)^{\frac{\pi}{2}}{^{{- {\alpha}}\; t}.}}}\end{matrix} & (173)\end{matrix}$

Combining S_(+0,−2)(t) of Eq. 25 and S*_(+0,−2) (t) of Eq. 173 oneobtains

$\begin{matrix}{{T_{{+ 0},{- 2}}\left( {t,M} \right)} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{- 2},K}}} \right)^{\; M_{K}\alpha_{K}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{- 2},K}}} \right)^{{- }\; M_{K}\alpha_{K}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{- 2},K}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}}} \right)} \\{^{\; M_{K - 1}\alpha_{K - 1}t} +} \\\begin{matrix}\begin{matrix}{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}}} \right)} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}}} \right)} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} -}\end{matrix} \\{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}}} \right)} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)^{\; M_{0}\alpha_{0}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}}} \right)^{{- }\; M_{0}\alpha_{0}t}} -}\end{matrix} \\{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{- 2},0}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}.}}} & (174)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c⁻⁰,c₊₂)-Sampling (PMS)

With S_(−0,+2)(t) of Eq. 29, the complex conjugate is proportional to

$\begin{matrix}\begin{matrix}{{{S_{{- 0} + 2}^{*}(t)} \propto {Q^{*}D_{{- 0},{+ 2}}{C_{{- 0},{+ 2}}(t)}}} = {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\begin{bmatrix}{c_{- 0}(t)} \\{c_{+ 2}(t)}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}c_{- 0} \\c_{+ 2}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}{\cos \left( {{\alpha \; t} - \Phi_{- 0}} \right)} \\{- {\sin \left( {{\alpha \; t} + \Phi_{+ 2}} \right)}}\end{bmatrix}}} \\{= {\begin{pmatrix}{{\cos \; \Phi_{- 0}{\cos \left( {\alpha \; t} \right)}} +} \\{\sin \; \Phi_{- 0}{\sin \left( {\alpha \; t} \right)}}\end{pmatrix} -}} \\{{\begin{pmatrix}{{\sin \; \Phi_{+ 2}{\cos \left( {\alpha \; t} \right)}} +} \\{\cos \; \Phi_{+ 2}{\sin \left( {\alpha \; t} \right)}}\end{pmatrix}}} \\{= {{\left( {{\cos \; \Phi_{- 0}} - {sin\Phi}_{+ 2}} \right){\cos \left( {\alpha \; t} \right)}} +}} \\{{\left( {{\sin \; \Phi_{- 0}} - {cos\Phi}_{+ 2}} \right){\sin \left( {\alpha \; t} \right)}}} \\{= {{\left( {{\cos \; \Phi_{- 0}} - {sin\Phi}_{+ 2}} \right)\frac{^{{\alpha}\; t} + ^{{- {\alpha}}\; t}}{2}} +}} \\{{\left( {{\sin \; \Phi_{- 0}} - {cos\Phi}_{+ 2}} \right)\frac{^{{\alpha}\; t} - ^{{- {\alpha}}\; t}}{2}}} \\{= {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}} \right)^{{\alpha}\; t}} -}} \\{{{\frac{}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}} \right)^{{\alpha}\; t}} +}} \\{{{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}} \right)^{{- {\alpha}}\; t}} +}} \\{{\frac{}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}} \right)^{{- {\alpha}}\; t}}} \\{= {{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}} \right)^{{\alpha}\; t}} -}} \\{{{\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}} \right)^{\frac{\pi}{2}}^{{\alpha}\; t}} +}} \\{{{\frac{1}{2}\left( {{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}} \right)^{{- {\alpha}}\; t}} +}} \\{{\frac{1}{2}\left( {{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}} \right)^{\frac{\pi}{2}}{^{{- {\alpha}}\; t}.}}}\end{matrix} & (175)\end{matrix}$

Combining S_(−0,+2)(t) of Eq. 29 and S*_(−0,+2) (t) of Eq. 175 oneobtains

$\begin{matrix}{{T_{{- 0},{+ 2}}\left( {t,M} \right)} = {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{+ 2},K}}} \right)^{\; M_{K}\alpha_{K}t}} -} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{- 0},K}} - {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{+ 2},K}}} \right)^{{- }\; M_{K}\alpha_{K}t}} +}\end{matrix} \\{\frac{M_{K}}{2}\left( {{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{+ 2},K}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)} \\{^{\; M_{K - 1}\alpha_{K - 1}t} -} \\\begin{matrix}\begin{matrix}{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +} \\{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}} \right)} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} +}\end{matrix} \\{\frac{M_{K - 1}}{2}\left( {{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}} \right)} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix} \otimes \ldots \otimes {\begin{pmatrix}{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{+ 2},0}}} \right)^{\; M_{0}\alpha_{0}t}} -} \\\begin{matrix}\begin{matrix}{{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} +} \\{{\frac{1}{2}\left( {{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{+ 2},0}}} \right)^{{- }\; M_{0}\alpha_{0}t}} +}\end{matrix} \\{\frac{M_{0}}{2}\left( {{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{+ 2},0}}} \right)^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}.}}} & (176)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

(c₊₀,c⁻²,c⁻⁰,c₊₂)-Sampling (DPMS)

As for Eq. 32, addition of S*_(+0,−2)(t) and S*_(−0,+2) (t) yields

$\begin{matrix}{{S_{{+ 0},{- 2},{- 0},{+ 2}}^{*}(t)} = {{{S_{{+ 0},{- 2}}^{*}(t)} + {S_{{- 0},{+ 2}}^{*}(t)}} \propto {{{- \frac{1}{2}}\begin{pmatrix}{{\cos \; \Phi_{+ 0}} - {\cos \; \Phi_{- 2}} +} \\{{\cos \; \Phi_{- 0}} - {\cos \; \Phi_{+ 2}}}\end{pmatrix}^{{\alpha}\; t}} + {\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{+ 0}} + {\sin \; \Phi_{- 2}} -} \\{{\sin \; \Phi_{- 0}} - {\sin \; \Phi_{+ 2}}}\end{pmatrix}^{\frac{\pi}{2}}^{{\alpha}\; t}} + {\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{+ 0}} + {\cos \; \Phi_{- 2}} +} \\{{\cos \; \Phi_{- 0}} + {\cos \; \Phi_{+ 2}}}\end{pmatrix}^{{- {\alpha}}\; t}} - {\frac{1}{2}\begin{pmatrix}{{\sin \; \Phi_{+ 0}} - {\sin \; \Phi_{- 2}} -} \\{{\sin \; \Phi_{- 0}} + {\sin \; \Phi_{+ 2}}}\end{pmatrix}^{\frac{\pi}{2}}{^{{- {\alpha}}\; t}.}}}}} & (177)\end{matrix}$

Combining S_(+0,−2,−0,+2)(t) of Eq. 32 and S*_(+0,−2,−0, +2)(t) of Eq.177 one obtains

$\begin{matrix}{{{T_{{+ 0},{- 2},{- 0},{+ 2}}(t)} \propto {\begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} + {\cos \; \Phi_{{- 2},K}} +} \\{{\cos \; \Phi_{{- 0},K}} + {\cos \; \Phi_{{+ 2},K}}}\end{pmatrix}^{\; M_{K}\alpha_{K}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} - {\sin \; \Phi_{{- 2},K}} -} \\{{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{+ 2},K}}}\end{pmatrix}^{\frac{\pi}{2}}^{\; M_{K}\alpha_{K}t}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},K}} - {\cos \; \Phi_{{- 2},K}} +} \\{{\cos \; \Phi_{{- 0},K}} - {\cos \; \Phi_{{+ 2},K}}}\end{pmatrix}^{{- }\; M_{K}\alpha_{K}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},K}} + {\sin \; \Phi_{{- 2},K}} -} \\{{\sin \; \Phi_{{- 0},K}} + {\sin \; \Phi_{{+ 2},K}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{K}\alpha_{K}t}}\end{matrix}\end{pmatrix} \otimes \begin{pmatrix}{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} + {\cos \; \Phi_{{- 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} + {\cos \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}} \\{^{\; M_{K - 1}\alpha_{K - 1}t} +} \\\begin{matrix}\begin{matrix}{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} - {\sin \; \Phi_{{- 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} + {\sin \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}} \\{{^{\frac{\pi}{2}}^{\; M_{K - 1}\alpha_{K - 1}t}} +} \\{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},{K - 1}}} - {\cos \; \Phi_{{- 2},{K - 1}}} +} \\{{\cos \; \Phi_{{- 0},{K - 1}}} - {\cos \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}} \\{^{{- }\; M_{K - 1}\alpha_{K - 1}t} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},{K - 1}}} + {\sin \; \Phi_{{- 2},{K - 1}}} -} \\{{\sin \; \Phi_{{- 0},{K - 1}}} - {\sin \; \Phi_{{+ 2},{K - 1}}}}\end{pmatrix}} \\{^{\frac{\pi}{2}}^{{- }\; M_{K - 1}\alpha_{K - 1}t}}\end{matrix}\end{pmatrix} \otimes \ldots \otimes \begin{pmatrix}{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} + {\cos \; \Phi_{{- 2},0}} +} \\{{\cos \; \Phi_{{- 0},0}} + {\cos \; \Phi_{{+ 2},0}}}\end{pmatrix}^{\; M_{0}\alpha_{0}t}} +} \\\begin{matrix}\begin{matrix}{{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} - {\sin \; \Phi_{{- 2},0}} -} \\{{\sin \; \Phi_{{- 0},0}} + {\sin \; \Phi_{{+ 2},0}}}\end{pmatrix}^{\frac{\pi}{2}}^{\; M_{0}\alpha_{0}t}} +} \\{{\frac{1}{2}\begin{pmatrix}{{\cos \; \Phi_{{+ 0},0}} - {\cos \; \Phi_{{- 2},0}} +} \\{{\cos \; \Phi_{{- 0},0}} - {\cos \; \Phi_{{+ 2},0}}}\end{pmatrix}^{{- }\; M_{0}\alpha_{0}t}} -}\end{matrix} \\{\frac{M_{K}}{2}\begin{pmatrix}{{\sin \; \Phi_{{+ 0},0}} + {\sin \; \Phi_{{- 2},0}} -} \\{{\sin \; \Phi_{{- 0},0}} - {\sin \; \Phi_{{+ 2},0}}}\end{pmatrix}^{\frac{\pi}{2}}^{{- }\; M_{0}\alpha_{0}t}}\end{matrix}\end{pmatrix}}},} & (178)\end{matrix}$

FT along the GFT dimension reveals mixed phase peak components at thedesired linear combination of chemical shifts, as well as at itsquadrature position. It also reveals ‘cross talk’ between GFT NMRsub-spectra, that is, the resulting peaks are located at linearcombinations of chemical shifts other than the desired one.

Combination of Different Sampling Schemes

Another aspect of the present invention relates to the generalization ofPMS and DPMS schemes for multiple indirect evolution times. In thesections above the discussion is based on the assumption that the samesampling scheme is employed for all evolution periods. Here it is shownthat it is readily possible to choose different sampling scheme for eachof the evolution periods.

With (c_(p),c_(q),c_(r),c_(s))-sampling scheme being employed for thej^(th) dimension of an (K+1)D experiment one has

S_(p,q,r,s)(t_(j))∝QD_(p,q,r,s)C_(p,q,r,s)(t_(j))  (179),

where {[p,q][r,s]≡[+0,+2][−0,−2]} for ‘States’-sampling,{[p,q][r,s]≡[+1,−1][+3,−3]} for π/4,π/4-shifted mirrored sampling and{[p,q][r,s]≡[+0,−2][−0,+2]} for 0,π/2-shifted mirrored sampling. Thesignal detected in the multidimensional NMR experiment is thenproportional to

$\begin{matrix}\begin{matrix}{{S\left( {t_{K},t_{K - 1},\ldots \mspace{14mu},t_{0}} \right)} = {{\overset{K}{\underset{j = 0}{\otimes}}{S_{p,q,r,s}\left( t_{j} \right)}} \propto {\overset{K}{\underset{j = 0}{\otimes}}{Q\overset{K}{\underset{j = 0}{\otimes}}D_{p,q,r,s}\underset{j = 0}{\overset{K}{\otimes}}{C_{p,q,r,s}\left( t_{j} \right)}}}}} \\{{= {{Q(K)}{D(K)}{C\left( {{t_{K,}t_{K - 1}},\ldots \mspace{14mu},t_{0}} \right)}}},}\end{matrix} & (180)\end{matrix}$

and one obtains for (K+1) jointly sampled chemical shifts in GFT NMR

$\begin{matrix}{{{{T_{p,q,r,s}(t)} \propto {\overset{K}{\underset{j = 0}{\otimes}}{G\overset{K}{\underset{j = 0}{\otimes}}D_{p,q,r,s}\overset{K}{\underset{j = 0}{\otimes}}{C_{p,q,r,s}(t)}}}} = {{G(K)}{D(K)}{C(t)}}}{{where},{t = {t_{0} = {{t_{1,}/\kappa_{1}} = {{t_{2}/\kappa_{2}} = {\ldots = {t_{3}/{\kappa_{3}.}}}}}}}}} & (181)\end{matrix}$

Inspection of Eqs. 180 and 181 shows that with the choice of aparticular sampling scheme for each shift evolution period, theresulting D matrix and C vector are constructed by straightforwardtensor product formation.

Applications in NMR Spectroscopy

Another aspect of the present invention relates to general applicationsof acquisition schemes in NMR spectroscopy.

Measurement of NMR Parameters

With Eqs. 4, 8, 15, 19, 26 and 30, one can calculate the secondary phaseshifts Φ_(±n) from the elements of the coefficient vector λ_(±n). Hence,one can design NMR experiments in which a parameter β, for exampleassociated with a short-lived nuclear spin state, is encoded into thesecondary phase shift. Then, measurement of the coefficient vectorλ_(±n) enables measurement of β. Parameter β can also be, for example, anuclear spin-spin coupling.

Optimization and (re-)Design of Radio-Frequency (r.f.) Pulse Schemes

Imperfections of r.f. pulse schemes introduce secondary phase shiftswhich are then considered to be ‘phase errors’. Optimization of r.f.pulse schemes requires minimization of phase errors to obtain pureabsorption mode NMR spectra. Measurement of the coefficient vector λenables measurement of phase errors. In turn, this allows one to derivehypotheses regarding the origin of these phase errors and corresponding(re-)design of the pulse sequence.

Concatenation of Two Step Phase Cycle for Solvent/Axial Peak Suppressionwith Dual ‘States’ and DPMS

One embodiment of the present invention involves concatenation of twostep phase cycle for solvent/axial peak suppression with dual ‘States’and DPMS

The residual solvent and axial peaks are not frequency labeled in allindirect dimensions. This feature enables one to concatenate dual‘States’ or DPMS acquisition with solvent/axial peak cancellation.

Hence, it is assumed in the following that the detected solvent/axialpeak time domain signal is independent of indirect evolution periods.

Dual ‘States’ Sampling

The solvent/axial peak signal is detected for each of c₊₀ and c₊₂interferograms. After transformation with D matrix and Q vector given inEq. 3, one thus obtains for the solvent signal:

$\begin{matrix}{{{{S_{{- 0},{- 2}}^{w}(t)} \propto {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\begin{bmatrix}{{I_{w}(t)}^{{\alpha}_{w}t_{aq}}} \\{{I_{w}(t)}^{{\alpha}_{w}t_{aq}}}\end{bmatrix}}} = {\left( {1 - } \right){I_{w}(t)}^{{\alpha}_{w}t_{aq}}}},} & (182)\end{matrix}$

where, I_(w),α_(w) and t_(aq) represent, respectively, the amplitude andfrequency of the time domain solvent/axial peak signal and the directacquisition time.

The c⁻⁰-interferogram can be acquired by shifting both the first 90° r.fpulse generating transverse magnetization for frequency labeling and thereceiver phase by π, so that the sign of the desired signal remainsunchanged. Hence, after transformations with D matrix and Q vector givenin Eq. 7, one obtains:

$\begin{matrix}{{{S_{{- 0},{- 2}}^{w}(t)} \propto {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{{- {I_{w}(t)}}^{{\alpha}_{w}t_{aq}}} \\{{I_{w}(t)}^{{\alpha}_{w}t_{aq}}}\end{bmatrix}}} = {{- \left( {1 - } \right)}{I_{w}(t)}{^{{\alpha}_{w}t_{aq}}.}}} & (183)\end{matrix}$

Addition of S^(w) _(+0,+2)(t_(aq)) in Eq. 182 and S^(w) _(−0,−2)(t_(aq))in Eq. 183 then results in cancelation of the residual solvent/axialpeak signal.

π/4 and 3π/4-Shifted Mirrored Sampling

For c₊₁ and c⁻¹ interferograms, one obtains after the transformationswith D matrix and Q vector given in Eq. 14:

$\begin{matrix}{{{S_{{+ 1},{- 1}}^{w}(t)} \propto {{{\frac{1}{\sqrt{2}}\left\lbrack {1\mspace{14mu} } \right\rbrack}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}} \\{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}}\end{bmatrix}}} = {\sqrt{2}{I_{w}(t)}{^{\; \alpha_{w}t_{aq}}.}}} & (184)\end{matrix}$

For the corresponding C₊₃ and c⁻³ interferograms one obtains aftertransformations with D matrix and Q vector given in Eq. 18:

$\begin{matrix}{{{S_{{+ 3},{- 3}}^{w}(t)} \propto {{{\frac{1}{\sqrt{2}}\left\lbrack {1\mspace{14mu} } \right\rbrack}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}} \\{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}}\end{bmatrix}}} = {{- \sqrt{2}}{I_{w}(t)}{^{\; \alpha_{w}t_{aq}}.}}} & (185)\end{matrix}$

Addition of S^(w) _(+1,−1)(t_(aq)) in Eq. 184 and S^(w) _(+3,−3)(t_(aq))in Eq. 185 results in cancellation of the residual solvent/axial peaksignal and r.f pulse or receiver phase shifts are not required.

0 and π/2-Shifted Mirrored Sampling

For c₊₀ and c⁻² interferograms, one obtains after the transformationswith D matrix and Q vector given in Eq. 25:

$\begin{matrix}{{{S_{{+ 0},{- 2}}^{w}(t)} \propto {{\left\lbrack {1\mspace{14mu} } \right\rbrack \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}} \\{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}}\end{bmatrix}}} = {\left( {1 + } \right){I_{w}(t)}{^{\; \alpha_{w}t_{aq}}.}}} & (186)\end{matrix}$

The c⁻⁰-interferogram can be acquired by shifting both the first 90° r.fpulse generating transverse magnetization for frequency labeling and thereceiver phase by π. Hence, after transformations with D matrix and Qvector given in Eq. 29, one obtains:

$\begin{matrix}{{{S_{{- 0},{+ 2}}^{w}(t)} \propto {{\left\lbrack {1\mspace{14mu} } \right\rbrack \begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\begin{bmatrix}{{- {I_{w}(t)}}^{\; \alpha_{w}t_{aq}}} \\{{I_{w}(t)}^{\; \alpha_{w}t_{aq}}}\end{bmatrix}}} = {{- \left( {1 + } \right)}{I_{w}(t)}{^{\; \alpha_{w}t_{aq}}.}}} & (187)\end{matrix}$

Addition of S^(w) _(+0,−2)(t_(aq)) in Eq. 186 and S^(w) _(−0,+2)(t_(aq))in Eq. 187 results in cancellation of the residual solvent/axial peaksignal.

Shift in Peak Position and Reduction of Peak Intensity as a Function ofPhase Error

Another aspect of the present invention relates to a shift in peakposition and reduction of peak intensity as a function of phase error.

With a time domain signal S(t) given by

S(t)=exp(iαt)*exp(−R ₂ t)*exp(iΦ)  (188),

where α denotes the chemical shift, R₂ represents the transverserelaxation rate constant and Φ is the phase error, one obtains (Cavanaghet al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press(2007), which is hereby incorporated by reference in its entirety) afterFT for the real part

Re(F [S(t)])=A(ω) cos Φ−D(ω) sin Φ  (189),

with A(ω)=R₂/[R₂ ²+(α−ω)²] and D(ω)=(α−ω)/[R₂ ²+(α−ω)²] representing theabsorptive and dispersive peak components. For a given Φ, the signalmaximum, I₀ ^(States)(Φ) and its position ω′ can be calculated bysolving

$\begin{matrix}{{{\frac{}{\omega}\left( {{{A(\omega)}\cos \; \Phi} - {{D(\omega)}\sin \; \Phi}} \right)} = 0},{{which}\mspace{14mu} {yields}}} & (190) \\{{\omega^{\prime} = {{\alpha - {R_{2}\left( \frac{{\cos \; \Phi} - 1}{\sin \; \Phi} \right)}} = {{\alpha - {\frac{\omega_{FWHH}}{2}\left( \frac{{\cos \; \Phi} - 1}{\sin \; \Phi} \right)}} = {{tg}^{2}\frac{\Phi}{2}}}}}{or}{{v^{\prime} = {{v_{0} - {\frac{v_{FWHH}}{2}\left( \frac{{\cos \; \Phi} - 1}{\sin \; \Phi} \right)}} = {{tg}^{2}\frac{\Phi}{2}}}},}} & (191)\end{matrix}$

where ν₀=α/2π and 2R₂=ω_(FWHH)=2πν_(FWHH) denotes the full width at halfheight. With ν_(FWHH)˜140 Hz and Φ=±15°, the shift α−ω′ in the positionof the maximum according to Eq. 191 thus amounts to be ˜10 Hz.

Substituting into the expression of A(ω) the value of ω′ one obtains themaximum intensity of a mixed phase signal with phase error Φ:

$\begin{matrix}\begin{matrix}{{I_{0}^{States}(\Phi)} = {{\cos \; \Phi \frac{R_{2}}{R_{2}^{2} + {R_{2}^{2}\left( \frac{{\cos \; \Phi} - 1}{\sin \; \Phi} \right)}^{2}}} -}} \\{{\sin \; \Phi \frac{R_{2}\left( \frac{{\cos \; \Phi} - 1}{\sin \; \Phi} \right)}{R_{2}^{2} + {R_{2}^{2}\left( \frac{{\cos \; \Phi} - 1}{\sin \; \Phi} \right)}^{2}}}} \\{= \frac{\sin^{2}\; \Phi}{2{R_{2}\left( {1 - {\cos \; \Phi}} \right)}}} \\{= {\frac{\cos^{2}\frac{\Phi}{2}}{R_{2}}.}}\end{matrix} & (192)\end{matrix}$

The maximum intensity, I₀ ^(States)(Φ=0), of an absorptive Lorentzianline is calculated by substituting ω=α in the expression of A(ω) givenabove, yielding I₀ ^(States)(Φ=0)=1/R₂. Hence, the ratio of theintensities of the signal with phase error and the absorptiveLorenstzian line is given by

$\begin{matrix} & (193)\end{matrix}$

With Eq. 80, the ratio of the intensities of the clean absorption modesignal obtained by (c₊₁,c⁻¹)-PMS, and the absorptive Lorentzian signalis given by

PMS = I 0 States  ( 0 )  cos   Φ I 0 States  ( 0 ) = cos   Φ . (194 )

FIG. 2 shows the reduction (in %) of the signal maximum, as (1−

)×100 for both ‘States’ and (c⁻¹,c⁻¹)-PMS versus Φ.

Implementation and Product Operator Formalism of Mirrored Sampling for‘Non-Constant Time’ Evolution Periods

Another aspect of the present invention relates to implementation andproduct operator formalism of mirrored sampling for ‘non-constant time’evolution periods.

States' Forward and Backward Sampling for Obtaining ¹³CFrequency-Labeled Magnetization (e.g. 2D [¹³C,¹H]-HSQC)

A product operator description (Ernst et al., “Principles of NuclearMagnetic Resonance in One and Two Dimensions,” Oxford: Oxford UniversityPress (1987); Jacobsen, N. E., “NMR Spectroscopy Explained,” Wiley, NewYork (2007), which are hereby incorporated by reference in theirentirety) for forward and backward sampling of ¹³C chemical shifts innon-constant time 2D [¹³C,¹H]-HSQC (Cavanagh et al., “Protein NMRSpectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which ishereby incorporated by reference in its entirety) (FIG. 3A-B) isprovided. The operators representing ¹³C and ¹H are denoted ‘C’ and ‘H’,respectively, and only terms yielding detected signal are retained.Before the first 90° pulse on ¹³C is applied, the density matrixσ(t_(A)) is proportional to (Cavanagh et al., “Protein NMRSpectroscopy,” 2nd Ed., San Diego Academic Press (2007), which is herebyincorporated by reference in its entirety) H_(z)C_(z), and afterfrequency-labeling the density matrix σ(t_(B)) is again proportional toH_(z)C_(z). Hence, in all cases the product operator description startsand ends with density matrices which are proportional to H_(z)C_(z),while they are modulated differently with the ¹³C shift as is requiredwhen considering the interferograms introduced in the section above.

Note that backward sampling requires the introduction of an additional180° pulse on ¹³C (FIGS. 3A-B). To ensure identical duty cycles forforward and backward sampling, a 180° pulse on ¹³C is also introducedfor forward sampling.

(c₊₀,c₊₂)-Sampling

The required interferograms are defined in Eq. 2.

c₊₀-Interferogram (FIG. 3A—Forward)

$\begin{matrix}{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{3\; {\pi/2}}}{\rightarrow}{{{- H_{z}}C_{x}}\overset{\alpha}{\rightarrow}{{{{- H_{z}}C_{x}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}}}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{{{- H_{z}}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}}}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {H_{z}C_{z}{{\cos \left( {\alpha \; t} \right)}.}}}}}}} & (195)\end{matrix}$

c₊₂-Interferogram (FIG. 3A—Forward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{0}}{\rightarrow}{{{- H_{z}}C_{y}}\overset{\alpha}{\rightarrow}{{{{- H_{z}}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}}}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {{- H_{z}}C_{z}{\sin \left( {\alpha \; t} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {{\alpha \; t} + \frac{\pi}{2}} \right)}.}}} & (196)\end{matrix}$

(c⁻⁰,c⁻²)-Sampling

The required interferograms are defined in Eq. 6.

c⁻⁰ —Interferogram (FIG. 3B—Backward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{3\; {\pi/2}}}{\rightarrow}{{{- H_{z}}C_{x}}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{{- H_{z}}C_{x}}\overset{\alpha}{\rightarrow}{{{{- H_{z}}C_{x}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {H_{z}C_{z}{\cos \left( {\alpha \; t} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {\alpha \; t} \right)}.}}} & (197)\end{matrix}$

c⁻²—Interferogram (FIG. 3B—Backward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{0}}{\rightarrow}{{{- H_{z}}C_{x}}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{H_{z}C_{y}}\overset{\alpha}{\rightarrow}{{{H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {H_{z}C_{z}{\sin \left( {\alpha \; t} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{2}} \right)}.}}} & (198)\end{matrix}$

π/4 and 3π/4-shifted mirrored sampling for obtaining ¹³Cfrequency-labeled magnetization (e.g. 2D [¹³C,¹H]-HSQC)

Note that for obtaining the c₊₁ and c⁻¹-interferograms, φ=−π/4, and forthe c₊₃ or c⁻³-interferograms, φ=π/4 (FIGS. 3A-B).

(c₊₁,c⁻¹)-Sampling (PMS)

The required interferograms are defined in Eq. 13.

c₊₁—Interferogram (FIG. 3A—Forward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{{- \pi}/4}}{\rightarrow}{{{- \frac{1}{\sqrt{2}}}\left( {{H_{z}C_{x}} + {H_{z}C_{y}}} \right)}\overset{\alpha}{\rightarrow}{{{- \frac{1}{\sqrt{2}}}\left( {{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}} \right)}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{{- \frac{1}{\sqrt{2}}}\left( {{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}} - {H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}} \right)}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {H_{z}{C_{z}\left( {{\cos \left( {\alpha \; t} \right)} - {\sin \left( {\alpha \; t} \right)}} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {{\alpha \; t} + \frac{\pi}{2}} \right)}.}}} & (199)\end{matrix}$

c⁻¹—Interferogram (FIG. 3B—Backward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{{- \pi}/4}}{\rightarrow}{{{- \frac{1}{\sqrt{2}}}\left( {{H_{z}C_{x}} + {H_{z}C_{y}}} \right)}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{{- \frac{1}{\sqrt{2}}}\left( {{H_{z}C_{x}} - {H_{z}C_{y}}} \right)}\overset{\alpha}{\rightarrow}{{{- \frac{1}{\sqrt{2}}}\left( {{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}} - {H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}} \right)}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {\frac{1}{\sqrt{2}}H_{z}{C_{z}\left( {{\cos \left( {\alpha \; t} \right)} + {\sin \left( {\alpha \; t} \right)}} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{2}} \right)}.}}} & (200)\end{matrix}$

(c₊₃,c⁻³)-Sampling (PMS)

The required interferograms are defined in Eq. 17.

c₊₃—Interferogram (FIG. 3A—Forward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{\pi/4}}{\rightarrow}{{\frac{1}{\sqrt{2}}\left( {{H_{z}C_{x}} - {H_{z}C_{y}}} \right)}\overset{\alpha}{\rightarrow}{{\frac{1}{\sqrt{2}}\left( {{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}} - {H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}} \right)}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{\frac{1}{\sqrt{2}}\left( {{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}} \right)}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {{- \frac{1}{\sqrt{2}}}H_{z}{C_{z}\left( {{\cos \left( {\alpha \; t} \right)} + {\sin \left( {\alpha \; t} \right)}} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {{\alpha \; t} + \frac{3\; \pi}{4}} \right)}.}}} & (201)\end{matrix}$

c⁻³—Interferogram (FIG. 3B—Backward)

$\begin{matrix}{{{{\sigma \left( t_{A} \right)} \propto {H_{z}C_{z}}}\overset{{({90{^\circ}})}_{\pi/4}}{\rightarrow}{{\frac{1}{\sqrt{2}}\left( {{H_{z}C_{x}} - {H_{z}C_{y}}} \right)}\overset{{({180{^\circ}})}_{0}}{\rightarrow}{{\frac{1}{\sqrt{2}}\left( {{H_{z}C_{x}} + {H_{z}C_{y}}} \right)}\overset{\alpha}{\rightarrow}{{\frac{1}{\sqrt{2}}\left( {{H_{z}C_{x}{\cos \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\sin \left( {\alpha \; t} \right)}} + {H_{z}C_{y}{\cos \left( {\alpha \; t} \right)}} - {H_{z}C_{x}{\sin \left( {\alpha \; t} \right)}}} \right)}\overset{{({90{^\circ}})}_{\pi/2}}{\rightarrow}{{\sigma \left( t_{B} \right)} \propto {{- \frac{1}{\sqrt{2}}}H_{z}{C_{z}\left( {{\cos \left( {\alpha \; t} \right)} - {\sin \left( {\alpha \; t} \right)}} \right)}}}}}}} = {H_{z}C_{z}{{\cos \left( {{{- \alpha}\; t} + \frac{3\; \pi}{4}} \right)}.}}} & (202)\end{matrix}$

Time-Proportional Phase Incrementation (TPPI)

Inspection of Eq. (B2) shows that, in one embodiment, the cases{ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1, need to beexcluded to ensure phase-sensitive detection of chemical shifts (seeparagraph [0015]). However, for these excluded cases, phase sensitivedetection can be accomplished by use of time-proportional phaseincrementation (TPPI) of radio-frequency pulse or receiver phases, awell known art in the field to detect chemical shifts phase-sensitively(Ernst et al.

“Principles of Nuclear Magnetic Resonance in One and Two Dimensions,”Oxford:

Oxford University Press (1987), which is hereby incorporated byreference in its entirety).

Sensitivity Enhancement

Combined forward and backward time domain sampling as described hereincan be employed in conjunction with preservation of equivalent pathways(PEP), an approach commonly used to enhance the sensitivity of NMR dataacquisition (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., SanDiego: Academic Press (2007), which is hereby incorporated by referencein its entirety).

Transverse Relaxation-Optimized NMR Spectroscopy

Transverse relaxation optimized NMR spectroscopy (TROSY) relies onspin-state selective measurement of chemical shifts to enhancesensitivity (Pervushin et al., Proc. Natl. Acad. Sci. U.S.A.,94:12366-12371 (1997), which is hereby incorporated by reference in itsentirety). Combined forward and backward time domain sampling asdescribed herein can be employed for TROSY.

Simultaneous Phase Cycled NMR Spectroscopy

Simultaneous phase cycled NMR spectroscopy is a method of simultaneouslyconducting more than one step of a radiofrequency phase cycle in anuclear magnetic resonance experiment (U.S. Pat. No. 7,408,346 toSzyperski et al., which is hereby incorporated by reference in itsentirety). Combined forward and backward time domain sampling asdescribed herein can be employed for simultaneous phase cycled NMRspectroscopy.

P- and N-Type Time Domain Data Acquisition

Phase sensitive detection can be accomplished by use of pulsed magneticfield gradients such that the amplitude modulation encoded in theC-vectors is encoded in the signal phase instead, yielding P- and N-typetime domain (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., SanDiego: Academic Press (2007), which is hereby incorporated by referencein its entirety). Linear combination of P- and N-type time domain yieldsthe amplitude modulated C-vectors. Hence, such P- and N-type time domaindata acquisition can be readily combined with combined forward andbackward time domain sampling as described herein.

Transformation by Use of Operator

The inventions described herein make use of transforming time domaininto frequency domain. Routinely, this is accomplished by use of FourierTransformation represented by operator F (Ernst et al., “Principles ofNuclear Magnetic Resonance in One and Two Dimensions,” Oxford: OxfordUniversity Press (1987), which is hereby incorporated by reference inits entirety) employed after multiplication of the C-vector withD-matrix and Q-vector. Since F is a linear operator, C-vectors canalternatively first be Fourier transformed into frequency domain andthen multiplied with D-matrix and Q-vector.

Other approaches relying on operators 0 with distinctly differentmathematical properties when compared with F have been established totransform time domain into frequency domain (Hoch et al. “NMR DataProcessing”, Wiley, New York (1996), which is hereby incorporated byreference in its entirety). For any approach relying on a linearoperator L, the operator can be employed as is described for F above.However, for non-linear operators (representing, for example, the‘maximum entropy method’), the transformation from time domain intofrequency domain is preferably, or necessarily, performed aftermultiplication of the C-vector with D-matrix and Q-vector in timedomain.

EXAMPLES

The Examples set forth below are for illustrative purposes only and arenot intended to limit, in any way, the scope of the present invention.

Example 1 Clean Absorption Mode NMR Data Acquisition for IdenticalSecondary Phase Shifts Data Acquisition

NMR spectra were acquired for ¹³C, ¹⁵N-labeled 8 kDa protein CaRl 78.All NMR spectra were acquired at 25° C. for CaR178, a target of theNortheast Structural Genomics Consortium (http://www.nesg.org), on aVarian INOVA 600 spectrometer equipped with a cryogenic¹H{¹³C,¹⁵N}-triple resonance probe. The ¹H carrier frequency was set tothe water line at 4.7 ppm throughout.

Non-constant time 2D [¹³C,¹H]-HSQC spectra comprising aliphatic signals(FIGS. 4A-I) were recorded along t₁(¹³C) and t₂(¹H), respectively, withspectral widths of 80.0 ppm and 13.4 ppm and t_(1,max)(¹³C)=42 ms andt_(2,max)(¹H)=64 ms. The ¹³C carrier frequency was set to 35.0 ppm.

Simultaneous constant time (Cavanagh et al., “Protein NMR Spectroscopy,”2nd Ed., San Diego: Academic Press (2007), which is hereby incorporatedby reference in its entirety) 2D [¹³C^(aliphatic)/¹³C^(aromatic)]-HSQCspectra comprising aliphatic and aromatic signals (FIG. 5) wererecorded, respectively, with spectral widths of 68.0 ppm and 13.4 ppmalong t₁(¹³C) and t₂(¹H) and t_(1,max)(¹³C)=21 ms and t_(2,max)(¹H)=64ms. The ¹³C carrier frequency was set to 43.0 ppm. Hence, aromaticsignals were folded once along ω₁(¹³C).

3D HC(C)H-TOCSY (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed.,San Diego: Academic Press (2007); Bax et al., J. Magn. Reson. 88:425-431(1990), which are hereby incorporated by reference in their entirety)spectra comprising aliphatic signals (FIGS. 6A-B) were recorded,respectively, with spectral widths of 13.4 ppm, 80.0 ppm and 13.4 ppmalong t₁(¹H), t₂(¹³C) and t₃(¹H), and t_(1,max)(¹H)=12 ms,t_(2,max)(¹³C)=11 ms and t_(3,max)(¹H)=64 ms. The ¹³C carrier frequencywas set to 36.0 ppm.

GFT (4,3)D C ^(αβ) C ^(β)(CO)NHN spectra (FIG. 7) were recorded,respectively, with spectral widths of 80.0 ppm, 32.0 ppm and 13.4 ppmalong t₁(¹³C^(α);¹³C^(αβ)), t₂(¹⁵N) and t₃(¹HN), andt_(1,max)(¹³C^(α);¹³C^(αβ))=7 ms, t_(2,max)(¹⁵N)=16 ms andt_(3,max)(¹HN)=64 ms. The ¹³C carrier frequency was set to 43.0 ppmduring the ¹³C^(αβ) shift evolution, and to 56.0 ppm during the ¹³C,shift evolution. The ¹⁵N carrier frequency was set at 118.0 ppm.

D and G matrix transformations were performed using programs written inC. Subsequently, spectra were processed and analyzed using the programsnmrPipe (Delaglio et al., J. Biomol. NMR, 6:277-293 (1995), which ishereby incorporated by reference in its entirety) and XEASY (Bartels etal., J. Biomol. NMR, 6:1-10 (1995), which is hereby incorporated byreference in its entirety). Prior to FT, time domain data weremultiplied by sine square bell window functions shifted by 75° andzero-filled at least twice. To ensure artifact-free signal comparison,no linear prediction (Cavanagh et al., “Protein NMR Spectroscopy,” 2ndEd., San Diego: Academic Press (2007), which is hereby incorporated byreference in its entirety) was applied.

Data Processing

In this section, the data processing of PMS and DPMS spectra presentedin Example 1 is described.

The interferograms

$c_{\pm n}:={\cos \left( {{{\pm \alpha}\; t} + \frac{n\; \pi}{4} + \Phi} \right)}$

were recorded and linearly combined (D matrix transformation) asdescribed above. D matrix transformation was followed by transformationwith either Q vector or G matrix (for GFT experiments). Finally, Fouriertransformation (FT) generated the desired frequency domain spectrum.

(c₊₁,c⁻¹)-PMS Simultaneous 2D [¹³C^(aliph)/¹³C^(arom),¹H]-HSQC

Time domain data corresponding to (c₊₁,c⁻¹)-PMS of t₁(¹³C) dimensionconsists of two 1D interferograms:

$\begin{matrix}{{S_{{+ 1},{- 1}}\left( t_{1} \right)} \propto {\begin{bmatrix}{c_{+ 1}\left( t_{1} \right)} \\{c_{- 1}\left( t_{1} \right)}\end{bmatrix}.}} & (203)\end{matrix}$

The two interferograms recorded for the experiment are listed in Table2.

TABLE 2 Interferograms recorded for t₁(¹³C)-(c₊₁, c⁻¹) simultaneous 2D[¹³C^(aliph)/¹³C^(arom),¹H]-HSQC Sampling of Modulation of detectedsignal Interferogram t₁(¹³C) (α₁ ≡ ¹³C) I1 c₊₁$\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)$I2 c⁻¹$\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)$With Eq. 92, for (c₊₁,c⁻¹)-PMS sampling D_(+1,−1) is given by

$\begin{matrix}{{D_{{+ 1},{- 1}} = \begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}},} & (204)\end{matrix}$

yielding, with the interferograms of Table 2, the linear combinations ofTable 3.

TABLE 3 Linearly combined interferograms after D matrix transformationCombined Modulation of detected signal interferograms Linear combination(α₁ ≡ ¹³C) L1 +I1 +I2${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}$L2 −I1 +I2${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}$

The linearly combined interferograms are then transformed with thevector Q=[1 i] to construct the spectrum comprising of complex timedomain signal as S_(+1,−1)(t)=L1+i*L2. Complex FT yields the desired thefrequency domain spectrum as shown in FIG. 5.

(c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS 3D HC(C)H TOCSY DPMS of more than one indirectdimension requires all combinations of PMS sampling as described in thesections above. The time domain data resulting from(c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS of two indirect dimensions consist of 16interferograms:

$\begin{matrix}\begin{matrix}{{\left\lfloor \begin{matrix}{{S_{{+ 1},{- 1}}\left( t_{2} \right)} +} \\{S_{{+ 3},{- 3}}\left( t_{2} \right)}\end{matrix} \right\rfloor \otimes \left\lfloor \begin{matrix}{{S_{{+ 1},{- 1}}\left( t_{1} \right)} +} \\{S_{{+ 3},{- 3}}\left( t_{1} \right)}\end{matrix} \right\rfloor} = {{{S_{{+ 1},{- 1}}\left( t_{2} \right)} \otimes {S_{{+ 1},{- 1}}\left( t_{1} \right)}} +}} \\{{{{S_{{+ 1},{- 1}}\left( t_{2} \right)} \otimes {S_{{+ 3},{- 3}}\left( t_{1} \right)}} +}} \\{{{{S_{{+ 3},{- 3}}\left( t_{2} \right)} \otimes {S_{{+ 1},{- 1}}\left( t_{1} \right)}} +}} \\{{{{S_{{+ 3},{- 3}}\left( t_{2} \right)} \otimes {S_{{+ 3},{- 3}}\left( t_{1} \right)}} \propto}} \\{{{\begin{bmatrix}{c_{+ 1}\left( t_{2} \right)} \\{c_{- 1}\left( t_{2} \right)}\end{bmatrix} \otimes \begin{bmatrix}{c_{+ 1}\left( t_{1} \right)} \\{c_{- 1}\left( t_{1} \right)}\end{bmatrix}} +}} \\{{{\begin{bmatrix}{c_{+ 1}\left( t_{2} \right)} \\{c_{- 1}\left( t_{2} \right)}\end{bmatrix} \otimes \begin{bmatrix}{c_{+ 3}\left( t_{1} \right)} \\{c_{- 3}\left( t_{1} \right)}\end{bmatrix}} +}} \\{{{\begin{bmatrix}{c_{+ 3}\left( t_{2} \right)} \\{c_{- 3}\left( t_{2} \right)}\end{bmatrix} \otimes \begin{bmatrix}{c_{+ 1}\left( t_{1} \right)} \\{c_{- 1}\left( t_{1} \right)}\end{bmatrix}} +}} \\{{\begin{bmatrix}{c_{+ 3}\left( t_{2} \right)} \\{c_{- 3}\left( t_{2} \right)}\end{bmatrix} \otimes \begin{bmatrix}{c_{+ 3}\left( t_{1} \right)} \\{c_{- 3}\left( t_{1} \right)}\end{bmatrix}}} \\{= {\begin{bmatrix}{{c_{+ 1}\left( t_{2} \right)}{c_{+ 1}\left( t_{1} \right)}} \\\begin{matrix}{{c_{+ 1}\left( t_{2} \right)}{c_{- 1}\left( t_{1} \right)}} \\{{c_{- 1}\left( t_{2} \right)}{c_{+ 1}\left( t_{1} \right)}} \\{{c_{- 1}\left( t_{2} \right)}{c_{- 1}\left( t_{1} \right)}}\end{matrix}\end{bmatrix} +}} \\{{\begin{bmatrix}{{c_{+ 1}\left( t_{2} \right)}{c_{+ 3}\left( t_{1} \right)}} \\\begin{matrix}{{c_{+ 1}\left( t_{2} \right)}{c_{- 3}\left( t_{1} \right)}} \\{{c_{- 1}\left( t_{2} \right)}{c_{+ 3}\left( t_{1} \right)}} \\{{c_{- 1}\left( t_{2} \right)}{c_{- 3}\left( t_{1} \right)}}\end{matrix}\end{bmatrix} +}} \\{{\begin{bmatrix}{{c_{+ 3}\left( t_{2} \right)}{c_{+ 1}\left( t_{1} \right)}} \\\begin{matrix}{{c_{+ 3}\left( t_{2} \right)}{c_{- 1}\left( t_{1} \right)}} \\{{c_{- 3}\left( t_{2} \right)}{c_{+ 1}\left( t_{1} \right)}} \\{{c_{- 3}\left( t_{2} \right)}{c_{- 1}\left( t_{1} \right)}}\end{matrix}\end{bmatrix} +}} \\{{\begin{bmatrix}{{c_{+ 3}\left( t_{2} \right)}{c_{+ 3}\left( t_{1} \right)}} \\\begin{matrix}{{c_{+ 3}\left( t_{2} \right)}{c_{- 3}\left( t_{1} \right)}} \\{{c_{- 3}\left( t_{2} \right)}{c_{+ 3}\left( t_{1} \right)}} \\{{c_{- 3}\left( t_{2} \right)}{c_{- 3}\left( t_{1} \right)}}\end{matrix}\end{bmatrix}.}}\end{matrix} & (205)\end{matrix}$

In (c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS 3D HC(C)H TOCSY, ¹H and ¹³C are sampled,respectively, in t₁ and t₂. The 16 interferograms of Eq. 205 wererecorded and are listed explicitly in Table 4. The interferograms of thefour terms in the last line of Eq. 205 correspond to the four sectionsseparated by dashed line in Table 4.

TABLE 4 2D interferograms recorded for (c₊₁, c⁻¹, c₊₃, c⁻³)-DPMS 3DHC(C)H TOCSY Sampling of Sampling of Modulation of detected signalInterferogram t₁(¹H) t₂(¹³C) (α₁ ≡ ¹H, α₂ ≡ ¹³C) Set A: I1 c₊₁ c₊₁${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I2 c₊₁ c⁻¹${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I3 c⁻¹ c₊₁${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I4 c⁻¹ c⁻¹${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$Set B: I5 c₊₁ c₊₃${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I6 c₊₁ c⁻³${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I7 c⁻¹ c₊₃${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I8 c⁻¹ c⁻³${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$Set C: I9 c₊₃ c₊₁${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I10 c₊₃ c⁻¹${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I11 c⁻³ c₊₁${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I12 c⁻³ c⁻¹${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{\pi}{4} + \Phi_{2}} \right)}$Set D: I13 c₊₃ c₊₃${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I14 c₊₃ c⁻³${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I15 c⁻³ c₊₃${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I16 c⁻³ c⁻³${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{2}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$

Set A of four 2D interferograms (I1 to I4) result from (c₊₁,c⁻¹)-PMS oftwo indirect dimensions. With Eq. 92, D_(t2,t1−(+1, −1)) for(c₊₁,c⁻¹)-PMS sampling of two indirect dimensions is given by

$\begin{matrix}\begin{matrix}{D_{{t\; 2},{{t\; 1} - {({{+ 1},{- 1}})}}} = {\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{t\; 2} \otimes \begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{t\; 1}}} \\{{= \begin{bmatrix}1 & 1 & 1 & 1 \\{- 1} & 1 & {- 1} & 1 \\{- 1} & {- 1} & 1 & 1 \\1 & {- 1} & {- 1} & 1\end{bmatrix}},}\end{matrix} & (206)\end{matrix}$

yielding, with set A of four interferograms of Table 4, linearcombinations L1-L4 of Table 5.

The linearly combined interferograms are then transformed by the vectorQ(1)=Q

Q=[1 i i −1] to construct the spectrum comprising complex time domainsignal S_(t) ₂ _(,t) ₁ −(1,−1)(t)=L1+i*L2+i*L3−L4. The frequency domainspectrum obtained after FT of the complex signal corresponds to Eq. 100.

Set B of four 2D interferograms (I15 to I8) result from (c₊₁,c⁻¹)-PMSfor t₁ (¹H) and (c₊₃,c⁻³)-PMS for t₂ (¹³C) and the corresponding Dmatrix transformation is given by:

$\begin{matrix}\begin{matrix}{D_{{{t\; 2} - {({{+ 3},{- 3}})}},{{t\; 1} - {({{+ 1},{- 1}})}}} = {\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{t\; 2} \otimes \begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{t\; 1}}} \\{{= \begin{bmatrix}{- 1} & {- 1} & {- 1} & {- 1} \\1 & {- 1} & 1 & {- 1} \\{- 1} & {- 1} & 1 & 1 \\1 & {- 1} & {- 1} & 1\end{bmatrix}},}\end{matrix} & (207)\end{matrix}$

yielding, with set B of four interferograms of Table 4, linearcombinations L5-L8 of Table 5.

The linearly combined interferograms are then transformed by the vectorQ(1)=Q

Q=[1 i i −1] to construct the spectrum comprising complex time domainsignal S_(t) ₂ _(−(3,−3), t) ₁ _(−(1,−1))(t)=L5+i*L6+i*L7−L8. Thefrequency domain spectrum obtained after FT of the complex signalcorresponds to Eq. 104.

Set C of four 2D interferograms (19 to 112) result from (c₊₃,c⁻³)—PMSfor t₁(¹H) and (c₊₁,c⁻¹)—PMS for t₂ (¹³C) and the corresponding D matrixtransformation is given by:

$\begin{matrix}\begin{matrix}{D_{{{t\; 2} - {({{+ 1},{- 1}})}},{{t\; 1} - {({{+ 3},{- 3}})}}} = {\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{t\; 2} \otimes \begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{t\; 1}}} \\{{= \begin{bmatrix}{- 1} & {- 1} & {- 1} & {- 1} \\{- 1} & 1 & {- 1} & 1 \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}},}\end{matrix} & (208)\end{matrix}$

yielding, with set C of four interferograms of Table 4, linearcombinations L9-L12 of Table 5.

The linearly combined interferograms are then transformed by the vectorQ(1)=Q

Q=[1 i i −1] to construct the spectrum comprising of complex time domainsignal S_(t) ₂ _(−(1,−1), t) ₁ _(−(3,−3))(t)=L9+i*L10+i*L11−L12. Thefrequency domain spectrum obtained after FT of the complex signalcorresponds to Eq. 108.

Set D of four 2D interferograms (I13 to I16) of Table 4 result from(c₊₃,c⁻³)-PMS for t₁ (¹H) and (C₊₃,c⁻³)-PMS for t₂ (¹³C) and thecorresponding D-matrix transformation is given by:

$\begin{matrix}\begin{matrix}{D_{{{t\; 2} - {({{+ 3},{- 3}})}},{{t\; 1} - {({{+ 3},{- 3}})}}} = {\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{t\; 2} \otimes \begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{t\; 1}}} \\{{= \begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}},}\end{matrix} & (209)\end{matrix}$

yielding, with set D of four interferograms of Table 4, linearcombinations L13-L16 of Table 5.

The linearly combined interferograms are then transformed by the vectorQ(1)=Q

Q=[1 i i −1] to construct the spectrum comprising complex time domainsignal S_(t) ₂ _(−(3,−3),t) ₁ _(−(3,−3))(t)=L13+i*L14+i*L15−L16. Thefrequency domain spectrum obtained after FT of the complex signalcorresponds to Eq. 111.

TABLE 5 Linearly combined interferograms after D matrix transformationCom- bined inter- fero- Modulation of detected signal grams Linearcombination (α₁ ≡ ¹H, α₂ ≡ ¹³C) L1 +I1 +I2 +I3 +I4${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L2 −I1 +I2 −I3 +I4${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L3 −I1 −I2 +I3 +I4${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L4 +I1 −I2 −I3 +I4${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L5 −I5 −I6 −I7 −I8${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L6 +I5 −I6 +I7 −I8${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L7 −I5 −I6 +I7 +I8${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L8 +I5 −I6 −I7 +I8${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L9 −I9 −I10 −I11 −I12${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L10 −I9 +I10 −I11 +I12${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L11 +I9 +I10 −I11 −I12${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L12 +I9 −I10 −I11 +I12${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L13 +I13 +I14 +I15 +I16${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L14 +I13 −I14 +I15 −I16${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$L15 +I13 +I14 −I15 −I16${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{2}} \right)}$L16 +I13 −I14 −I15 +I16${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{2}} \right)}$Addition of the four PMS frequency domain spectra resulting from sets Ato D in Table 5, yields cancellation of quad peaks so that solelyabsorptive peaks remain with intensities being proportional to cos Φ1cos Φ₂.

Alternatively, all linear combinations can be performed in the timedomain using Eq. 180, or in an explicit form:

$\begin{matrix}{{S \propto {\underset{j = 0}{\overset{1}{\otimes}}{Q\underset{j = 0}{\overset{1}{\otimes}}D_{({{+ 1},{- 1},{+ 3},{- 3}})}\underset{j = 0}{\overset{1}{\otimes}}{C_{({{+ 1},{- 1},{+ 3},{- 3}})}\left( t_{j} \right)}}}} = {\left\lbrack {{1\mspace{14mu} \mspace{14mu} }\mspace{14mu} - 1} \right\rbrack \cdot \begin{bmatrix}1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & 1 & 1 \\{- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} \\{- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1\end{bmatrix} \cdot {\begin{bmatrix}{I\; 1} \\{I\; 2} \\{I\; 3} \\{I\; 4} \\{I\; 5} \\{I\; 6} \\{I\; 7} \\{I\; 8} \\{I\; 9} \\{I\; 10} \\{I\; 11} \\{I\; 12} \\{I\; 13} \\{I\; 14} \\{I\; 15} \\{I\; 16}\end{bmatrix}.}}} & (210)\end{matrix}$

(c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS (4,3)D C ^(αβ) C ^(α)(CO)NHN

Processing of GFT spectra acquired with (c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS for twojointly measured chemical shifts is analogous to what is described for(c₊₁,c⁻¹,c₊₃,c⁻³)-DMPS 3D HC(C)H TOCSY, except that, prior to FT, a Gmatrix transformation replaces the Q vector transformation.

In the (c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS (4,3)D C ^(αβ) C ^(α)(CO)NHN, ¹³C^(α) and¹³C^(αβ) chemical shift evolutions are jointly sampled and the resultingtime domain data consist of 16 interferograms.

$\begin{matrix}\begin{matrix}{{\left\lfloor \begin{matrix}{{S_{C^{\alpha\beta} - {({{+ 1},{- 1}})}}\left( t_{1} \right)} +} \\{S_{C^{\alpha\beta} - {({{+ 3},{- 3}})}}\left( t_{1} \right)}\end{matrix} \right\rfloor \otimes \left\lfloor \begin{matrix}{{S_{C^{\alpha} - {({{+ 1},{- 1}})}}\left( t_{1} \right)} +} \\{S_{C^{\alpha} - {({{+ 3},{- 3}})}}\left( t_{1} \right)}\end{matrix} \right\rfloor} = {{S_{C^{\alpha\beta} - {({{+ 1},{- 1}})}}\left( t_{1} \right)} \otimes}} \\{{{S_{C^{\alpha} - {({{+ 1},{- 1}})}}\left( t_{1} \right)} +}} \\{{{S_{C^{\alpha\beta} - {({{+ 1},{- 1}})}}\left( t_{2} \right)} \otimes}} \\{{{S_{C^{\alpha} - {({{+ 3},{- 3}})}}\left( t_{1} \right)} +}} \\{{{S_{C^{\alpha\beta} - {({{+ 3},{- 3}})}}\left( t_{2} \right)} \otimes}} \\{{{S_{C^{\alpha} - {({{+ 1},{- 1}})}}\left( t_{1} \right)} +}} \\{{{S_{C^{\alpha\beta} - {({{+ 3},{- 3}})}}\left( t_{2} \right)} \otimes}} \\{{{S_{C^{\alpha} - {({{+ 3},{- 3}})}}\left( t_{1} \right)} \propto}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 1})}}\left( t_{1} \right)}\end{bmatrix} \otimes}} \\{{\begin{bmatrix}{c_{C^{\alpha} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 1})}}\left( t_{1} \right)}\end{bmatrix} +}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 1})}}\left( t_{1} \right)}\end{bmatrix} \otimes}} \\{{\begin{bmatrix}{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)}\end{bmatrix} +}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 3})}}\left( t_{1} \right)}\end{bmatrix} \otimes}} \\{{\begin{bmatrix}{c_{C^{\alpha} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 1})}}\left( t_{1} \right)}\end{bmatrix} +}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 3})}}\left( t_{1} \right)}\end{bmatrix} \otimes}} \\{\begin{bmatrix}{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 3})}}\left( t_{1} \right)}\end{bmatrix}} \\{= {\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 1})}}\left( t_{1} \right)} \\\begin{matrix}{c_{C^{\alpha} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 1})}}\left( t_{1} \right)}\end{matrix}\end{bmatrix} +}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 1})}}\left( t_{1} \right)} \\\begin{matrix}{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)}\end{matrix}\end{bmatrix} +}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 3})}}\left( t_{1} \right)} \\\begin{matrix}{c_{C^{\alpha} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 1})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({- 1})}}\left( t_{1} \right)}\end{matrix}\end{bmatrix} +}} \\{{\begin{bmatrix}{c_{C^{\alpha\beta} - {({+ 3})}}\left( t_{1} \right)} \\\begin{matrix}{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha\beta} - {({- 3})}}\left( t_{1} \right)} \\{c_{C^{\alpha} - {({+ 3})}}\left( t_{1} \right)}\end{matrix}\end{bmatrix}.}}\end{matrix} & (211)\end{matrix}$

The 16 interferograms of Eq. 211 were recorded and are listed explicitlyin Table 6. The interferograms of the four terms of Eq. 211 correspondto the four sections separated by dashed line in Table 6.

TABLE 6 2D interferograms recorded for DPMS (4,3)D C ^(αβ) C ^(α)(CO)NHNSampling of Sampling of Modulation of detected signal Interferogramt₁(¹³C^(α)) t₁(¹³C^(αβ)) (α₁ ≡ ¹³C^(α), α₂ ≡ ¹³C^(αβ)) Set A: I1 c₊₁ c₊₁${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I2 c₊₁ c⁻¹${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I3 c⁻¹ c₊₁${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I4 c⁻¹ c⁻¹${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$Set B: I5 c₊₁ c₊₃${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I6 c₊₁ c⁻³${\cos \left( {{\alpha_{1}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I7 c⁻¹ c₊₃${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I8 c⁻¹ c⁻³${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$Set C: I9 c₊₃ c₊₁${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I10 c₊₃ c⁻¹${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$I11 c⁻³ c₊₁${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t} + \frac{\pi}{4} + \Phi_{2}} \right)}$I12 c⁻³ c⁻¹${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{\pi}{4} + \Phi_{2}} \right)}$Set D: I13 c₊₃ c₊₃${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I14 c₊₃ c⁻³${\cos \left( {{\alpha_{1}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I15 c⁻³ c₊₃${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{\alpha_{2}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$I16 c⁻³ c⁻³${\cos \left( {{{- \alpha_{1}}t_{1}} + \frac{3\pi}{4} + \Phi_{1}} \right)}{\cos \left( {{{- \alpha_{2}}t_{1}} + \frac{3\pi}{4} + \Phi_{2}} \right)}$

Set A of four 2D interferograms (I1 to I4) result from (c₊₁,c⁻¹)-PMS oftwo jointly sampled indirect dimensions. D_(C) _(αβ) _(−(+1,−1)) for(c₊₁,c⁻¹)-PMS sampling of two indirect dimensions is given by

$\begin{matrix}\begin{matrix}{D_{C^{\alpha\beta},{C^{\alpha} - {({{+ 1},{- 1}})}}} = {\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{C^{\alpha\beta}} \otimes \begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{C^{\alpha}}}} \\{= {\begin{bmatrix}1 & 1 & 1 & 1 \\{- 1} & 1 & {- 1} & 1 \\{- 1} & {- 1} & 1 & 1 \\1 & {- 1} & {- 1} & 1\end{bmatrix}.}}\end{matrix} & (212)\end{matrix}$

yielding, with set A of four interferograms of Table 6, linearcombinations L1-L4 of Table 7.

The linearly combined interferograms are then transformed by the Gmatrix given by

$\begin{matrix}\begin{matrix}{{G(1)} = {\begin{bmatrix}Q \\Q^{*}\end{bmatrix} \otimes \lbrack Q\rbrack_{0}}} \\{= {\begin{bmatrix}1 &  \\1 & {- }\end{bmatrix}_{1} \otimes \left\lbrack {1\mspace{14mu} } \right\rbrack_{0}}} \\{= {\begin{bmatrix}1 &  &  & {- 1} \\1 &  & {- } & 1\end{bmatrix}.}}\end{matrix} & (213)\end{matrix}$

Hence, FT of L1+i*L2+i*L3−L4 gives rise to the sub-spectrum where α₁+α₂is observed (the second column of Table 8), while FT of L1+i*L2−i*L3+L4gives rise to the sub-spectrum where α₁-α₂ is observed (the secondcolumn of Table 9).

Set B of four 2D interferograms (I5 to I8) result from (c₊₁,c⁻¹)-PMS fort₁ (¹³C^(α)) and (c₊₃,c⁻³)-PMS for t₁ (¹³C^(αβ)) and the corresponding Dmatrix transformation is given by:

$\begin{matrix}\begin{matrix}{D_{{C^{\alpha\beta} - {({{+ 3},{- 3}})}},{C^{\alpha} - {({{+ 1},{- 1}})}}} = {\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{C^{\alpha\beta}} \otimes \begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{C^{\alpha}}}} \\{= {\begin{bmatrix}{- 1} & {- 1} & {- 1} & {- 1} \\1 & {- 1} & 1 & {- 1} \\{- 1} & {- 1} & 1 & 1 \\1 & {- 1} & {- 1} & 1\end{bmatrix}.}}\end{matrix} & (214)\end{matrix}$

yielding, with set B of four interferograms of Table 6, linearcombinations L5-L8 of Table 7.

The linearly combined interferograms are then transformed by the Gmatrix described in Eq. 213. Hence, FT of L5+i*L6+i*L7−L8 gives rise tothe sub-spectrum where α₁+α₂ is observed (the third column of Table 8),while FT of L5+i*L6−i*L7+L8 gives rise to the sub-spectrum where α₁-α₂is observed (the third column of Table 9).

Set C of four 2D interferograms (19 to 112) result from ((c₊₃,c⁻³)-PMSfor t₁ (¹³C^(α)) and (c₊₁,c⁻¹)-PMS for t₁ (¹³CO^(αβ)) and thecorresponding D matrix transformation is given by:

$\begin{matrix}\begin{matrix}{D_{{C^{\alpha\beta} - {({{+ 1},{- 1}})}},{C^{\alpha} - {({{+ 3},{- 3}})}}} = {\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}_{C^{\alpha\beta}} \otimes \begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{C^{\alpha}}}} \\{= {\begin{bmatrix}{- 1} & {- 1} & {- 1} & {- 1} \\{- 1} & 1 & {- 1} & 1 \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}.}}\end{matrix} & (215)\end{matrix}$

yielding, with set C of four interferograms of Table 6, linearcombinations L9-L12 of Table 7.

The linearly combined interferograms are then transformed by the Gmatrix described in Eq. 213. Hence, FT of L9+i*L10+i*L11−L12 gives riseto the sub-spectrum where α₁+α₂ is observed (the fourth column of Table8), while FT of L9+i*L10−i*L11+L12 gives rise to the sub-spectrum whereα₁−α₂ is observed (the fourth column of Table 9).

Set D of four 2D interferograms (I13 to I16) of Table 4 result from(c₊₃,c⁻³)-PMS for t₁(¹H) and (c₊₃,c⁻³)-PMS for t₂ (¹³C) and thecorresponding D matrix transformation is given by:

$\begin{matrix}\begin{matrix}{D_{{C^{\alpha\beta} - {({{+ 3},{- 3}})}},{C^{\alpha} - {({{+ 3},{- 3}})}}} = {\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{C^{\alpha\beta}} \otimes \begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}_{C^{\alpha}}}} \\{= {\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}.}}\end{matrix} & (216)\end{matrix}$

yielding, with set D of four interferograms of Table 6, linearcombinations L13-L16 of Table 7.

The linearly combined interferograms are then transformed by the Gmatrix described in Eq. 213. Hence, FT of L13+i*L14+i*L15−L16 gives riseto the sub-spectrum where α₁+α2 is observed (the last column of Table8), while FT of L13+i*L14−i*L15+L16 gives rise to the sub-spectrum whereα₁-α₂ is observed (the last column of Table 9).

TABLE 7 Linearly combined interferograms after D matrix transformationCom- bined inter- fero- Modulation of detected signal grams Linearcombination (α₁ ≡ ¹³C^(α), α₂ ≡ ¹³C^(αβ)) L1 +I1 +I2 +I3 +I4${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L2 −I1 +I2 −I3 +I4${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L3 −I1 −I2 +I3 +I4${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L4 +I1 −I2 −I3 +I4${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L5 −I5 −I6 −I7 −I8${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L6 +I5 −I6 +I7 −I8${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L7 −I5 −I6 +I7 +I8${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L8 +I5 −I6 −I7 +I8${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L9 −I9 −I10 −I11 −I12${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L10 −I9 +I10 −I11 +I12${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L11 +I9 +I10 −I11 −I12${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L12 +I9 −I10 −I11 +I12${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L13 +I13 +I14 +I15 +I16${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L14 +I13 −I14 +I15 −I16${\sin \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\cos \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$L15 +I13 +I14 −I15 −I16${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\sin \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\cos \left( {\alpha_{2}t_{1}} \right)}$L16 +I13 −I14 −I15 +I16${\cos \left( {\frac{\pi}{4} + \Phi_{1}} \right)}{\cos \left( {\frac{\pi}{4} + \Phi_{2}} \right)}{\sin \left( {\alpha_{1}t_{1}} \right)}{\sin \left( {\alpha_{2}t_{1}} \right)}$Table 8 and 9, respectively, list the peak intensities of thesub-spectrum measuring α₁+α₂ and sub-spectrum measuring α₁-α₂, for allcombinations of PMS given in Eq.211.

TABLE 8 Relative peaks intensities in the sub-spectrum measuring α₁ + α₂t₁(¹³C^(α)) − (c₊₁, c⁻¹), t₁(¹³C^(α)) − (c₊₁, c⁻¹), t₁(¹³C^(α)) − (c₊₃,c⁻³), t₁(¹³C^(α)) − (c₊₃, c⁻³), Peaks at t₁(¹³C^(αβ)) − (c₊₁, c⁻¹)t₁(¹³C^(αβ)) − (c₊₃, c⁻³) t₁(¹³C^(αβ)) − (c₊₁, c⁻¹) t₁(¹³C^(αβ)) − (c₊₃,c⁻³) α₁ + α₂ cosΦ₁cosΦ₂ cosΦ₁cosΦ₂ cosΦ₁cosΦ₂ cosΦ₁cosΦ₂ α₁ − α₂−cosΦ₁sinΦ₂ cosΦ₁sinΦ₂ −cosΦ₁sinΦ₂ cosΦ₁sinΦ₂ −α₁ + α₂ −sinΦ₁cosΦ₂−sinΦ₁cosΦ₂ sinΦ₁cosΦ₂ sinΦ₁cosΦ₂ −α₁ − α₂ sinΦ₁sinΦ₂ −sinΦ₁sinΦ₂−sinΦ₁sinΦ₂ sinΦ₁sinΦ₂Inspection of Table 8 shows that the cross-talk peak located at α₁−α₂ aswell as the quad peaks at −α₁+α₂ and −α₁−α₂, cancel when all differentlysampled sub-spectra are added up. Complex FT yields the desired thefrequency domain spectrum as shown in FIG. 7.

TABLE 9 Relative peaks intensities in the sub-spectrum measuring α₁ − α₂t₁(¹³C^(α)) − (c₊₁, c⁻¹), t₁(¹³C^(α)) − (c₊₁, c⁻¹), t₁(¹³C^(α)) − (c₊₃,c⁻³), t₁(¹³C^(α)) − (c₊₃, c⁻³), Peaks at t₁(¹³C^(αβ)) − (c₊₁, c⁻¹)t₁(¹³C^(αβ)) − (c₊₃, c⁻³) t₁(¹³C^(αβ)) − (c₊₁, c⁻¹) t₁(¹³C^(αβ)) − (c₊₃,c⁻³) α₁ − α₂ cosΦ₁cosΦ₂ cosΦ₁cosΦ₂ cosΦ₁cosΦ₂ cosΦ₁cosΦ₂ α₁ + α₂−cosΦ₁sinΦ₂ cosΦ₁sinΦ₂ −cosΦ₁sinΦ₂ cosΦ₁sinΦ₂ −α₁ − α₂ −sinΦ₁cosΦ₂−sinΦ₁cosΦ₂ sinΦ₁cosΦ₂ sinΦ₁cosΦ₂ −α₁ + α₂ sinΦ₁sinΦ₂ −sinΦ₁sinΦ₂−sinΦ₁sinΦ₂ sinΦ₁sinΦ₂Inspection of Table 9 shows that the cross-talk peak located at α₁+α₂ aswell as the quad peaks at −α₁+α₂ and −α₁—α₂, cancel when all differentlysampled sub-spectra are added up.

Alternatively, all linear combinations can be performed in time domainusing Eq. 182, or in an explicit form:

$\begin{matrix}{{T \propto {\underset{j = 0}{\overset{1}{\otimes}}{G\underset{j = 0}{\overset{1}{\otimes}}D_{({{+ 1},{- 1},{+ 3},{- 3}})}\underset{j = 0}{\overset{1}{\otimes}}{C_{({{+ 1},{- 1},{+ 3},{- 3}})}\left( t_{j} \right)}}}} = {{\begin{bmatrix}1 &  &  & {- 1} \\1 &  & {- } & 1\end{bmatrix} \cdot {\quad{\left\lbrack \begin{matrix}1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & 1 & 1 \\{- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} \\{- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1\end{matrix} \right\rbrack  \cdot}\quad}}{\quad{\quad{\quad{\quad {\quad{\quad {\quad {\quad{\quad {\quad{\quad{\begin{bmatrix}{I\; 1} \\{I\; 2} \\{I\; 3} \\{I\; 4} \\{I\; 5} \\{I\; 6} \\{I\; 7} \\{I\; 8} \\{I\; 9} \\{I\; 10} \\{I\; 11} \\{I\; 12} \\{I\; 13} \\{I\; 14} \\{I\; 15} \\{I\; 16}\end{bmatrix}.}}}}}}}}}}}}}} & (221)\end{matrix}$

Comparative Cross Sections

In order to demonstrate the salient peak detection features of thesampling schemes alluded to in the text, non-constant time 2D[¹³C,¹H]-HSQC spectra were recorded with delayed acquisition, that is,the first FID was acquired with t₁(¹³C)=25 μs (FIGS. 4A-I). Thisintroduces a large ‘phase error’ of 1.35°/ppm, with a total of 108°across the 80.0 ppm spectral width. Furthermore, 2D [¹³C, ¹H]-HSQC wasemployed to exemplify non-constant time forward and backward sampling.

Discussion

First, (C₊₁,c⁻¹)-, (c₊₀,c⁻²)-PMS, and corresponding DPMS was implementedand tested for 2D [¹³C,¹H]-HSQC (Cavanagh et al., “Protein NMRSpectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohret al., “Multidimensional Solid-State NMR and Polymers,” New York:Academic Press (1994), which are hereby incorporated by reference intheir entirety). The implementation of non-constant time (Cavanagh etal., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press(2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR andPolymers,” New

York: Academic Press (1994), which are hereby incorporated by referencein their entirety) ‘backward-sampling’ required introduction of anadditional 180° ¹³C radio-frequency (r.f) pulse (FIGS. 3A-B). PMS andDPMS remove dispersive components and yield clean absorption modespectra (FIGS. 4A-I) without a phase correction.

(c₊₁,c⁻¹)-PMS and (c₊₁,c⁻¹,c₊₃,c⁻³)-DPMS was then employed forsimultaneous constant-time 2D [¹³C^(aliphatic)/¹³C^(aromatic),¹H]-HSQCin which aromatic signals are folded. Since frequency labeling wasaccomplished in a constant-time manner, (Cavanagh et al., “Protein NMRSpectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohret al., “Multidimensional Solid-State NMR and Polymers,” New York:Academic Press (1994), which are hereby incorporated by reference intheir entirety) no r.f. pulses had to be added to the pulse scheme(Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego:Academic Press (2007); Schmidt-Rohr et al., “MultidimensionalSolid-State NMR and Polymers,” New York: Academic Press (1994), whichare hereby incorporated by reference in their entirety). The phaseerrors of the folded aromatic signals cannot be corrected afterconventional data acquisition, (Ernst et al., “Principles of NuclearMagnetic Resonance in One and Two Dimensions,” Oxford:

Oxford University Press (1987), which is hereby incorporated byreference in its entirety) but are eliminated with PMS (FIG. 5).

Multiple (c⁻¹,c⁻¹,c₊₃,c⁻³)-DPMS was exemplified for 3D HC(C)H totalcorrelation spectroscopy (TOCSY) (Bax et al., J. Magn. Reson. 88:425-431(1990), which is hereby incorporated by reference in its entirety). The¹³C-¹³C isotropic mixing introduces phase errors along ω₁(¹³C) whichcannot be entirely removed by prior art techniques. Moreover, inhetero-nuclear resolved NMR spectra comprising ¹H-¹³C planes withintense diagonal peaks [e.g. HC(C)H TOCSY], even small phase errorsimpede identification of cross peaks close to the diagonal. Since ¹H and¹³C frequency labeling was accomplished in a semi constant-time manner(Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego:Academic Press (2007); Schmidt-Rohr et al., “MultidimensionalSolid-State NMR and Polymers,” New York: Academic Press (1994), which ishereby incorporated by reference in its entirety), no r.f. pulses had tobe added to the pulse scheme (Bax et al., J. Magn. Reson. 88:425-431(1990)). Comparison with the conventionally acquired spectrum showselimination of dispersive components in both indirect dimensions (FIGS.6A-B).

To exemplify multiple (c⁻¹,c⁻¹,c₊₃,c⁻³)-DPMS for GFT NMR, (Kim et al.,J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad.Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476(2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yanget al., J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., MethodsEnzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A.102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692(2007), which are hereby incorporated by reference in their entirety) itwas employed for (4,3)D C ^(αβ) C ^(α)(CO)NHN (Atreya et al., Proc.Natl. Acad. Sci. USA 101:9642-9647 (2004), which is hereby incorporatedby reference in its entirety) in both the ¹³C^(αβ) and ¹³C^(α) shiftevolution periods. Since frequency labeling was accomplished in aconstant-time manner (Cavanagh et al., “Protein NMR Spectroscopy,” 2ndEd., San Diego: Academic Press (2007); Schmidt-Rohr et al.,“Multidimensional Solid-State NMR and Polymers,” New

York: Academic Press (1994); Atreya et al., Proc. Natl. Acad. Sci. USA101:9642-9647 (2004), which are hereby incorporated by reference intheir entirety), no r.f. pulses had to be added to the pulse scheme(Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004), whichis hereby incorporated by reference in its entirety). Comparison withstandard GFT NMR shows elimination of dispersive components in theGFT-dimension (FIG. 7). Importantly, only PMS can eliminate entirelydispersive components in GFT-based projection NMR (Kim et al., J. Am.Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci.USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004);Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al.,J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol.394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A.102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692(2007); Szyperski et al., Magn. Reson. Chem. 44:51-60 (2006); Kupce etal., J. Am. Chem. Soc. 126:6429-6440 (2004); Coggins et al., J. Am.Chem. Soc. 126:1000-1001 (2004); Eghbalnia et al., J. Am. Chem. Soc.127:12528-12536 (2005); Hiller et al., Proc. Natl. Acad. Sci. U.S.A.102:10876-10881 (2005), which are hereby incorporated by reference intheir entirety).

Dispersive components shift peak maxima. For example, in routinelyacquired (4,3)D C^(αβ)C^(α)(CO)NHN (Atreya et al., Proc. Natl. Acad.Sci. USA 101:9642-9647 (2004), which is hereby incorporated by referencein its entirety), signals exhibit full widths at half height ofΔν_(FWHH)˜140 Hz in the GFT dimension. Since phase errors up to about±15° are observed, maxima are shifted by up to about ±10 Hz (˜±0.07 ppmat 600 MHz ¹H resonance frequency) and the precision of chemical shiftmeasurements is reduced accordingly.

Example 2 Measurement of Secondary Phase Shifts Using PMS

In this section, the protocol for measurement of secondary phase shiftsfrom PMS spectra is described.

π/4 and 3π/4-Shifted Combined Forward-Backward Sampling

(c₊₁,c⁻¹)-Sampling (PMS)

The two interferograms for (c₊₁,c⁻¹)-PMS are given by Eq. 13 and theresulting complex time domain signal S_(+1,−1)(t), according to Eq. 14,is proportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 1},{- 1}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 1},{- 1}}{C_{{+ 1},{- 1}}(t)}}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}}} \\{{\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \frac{\pi}{4} + \Phi_{+ 1}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{\pi}{4} + \Phi_{- 1}} \right)}\end{bmatrix} =}} \\{= {\sqrt{2}{\cos \left( \frac{\Phi_{+ 1} + \Phi_{- 1}}{2} \right)}}} \\{{^{\lbrack{{{\alpha}\; t} + {\frac{\Phi_{+ 1} - \Phi_{- 1}}{2}}}\rbrack} -}} \\{{\sqrt{2}{\sin \left( \frac{\Phi_{+ 1} + \Phi_{- 1}}{2} \right)}}} \\{{^{\lbrack{{{\alpha}\; t} - {\frac{\Phi_{+ 1} - \Phi_{- 1}}{2}}}\rbrack}.}}\end{matrix} & (222)\end{matrix}$

FT reveals a superposition of absorptive and dispersive components atboth the actual and the quadrature peak positions, with peak intensitiesproportional to

$\cos \left( \frac{\Phi_{+ 1} + \Phi_{- 1}}{2} \right)$ and${\sin \left( \frac{\Phi_{+ 1} + \Phi_{- 1}}{2} \right)},$

respectively. The phase shifts are given by

$\frac{\Phi_{+ 1} - \Phi_{- 1}}{2}.$

In the following, a protocol is described for obtaining the secondaryphase shifts from experimentally measured peak volumes and the phaseshifts of actual and quad peaks.The protocol involves two steps:

Step 1:

Phase correct both the actual and the quad peaks, thereby measuring theassociated phase shifts

$\frac{\Phi_{+ 1} - \Phi_{- 1}}{2}.$

For brevity,

$\Phi_{diff} = \frac{\Phi_{+ 1} - \Phi_{- 1}}{2}$

in the following.

Step 2:

Perform frequency domain signal integration on the phase correctedactual and the quad peaks to obtain the peak ‘volumes’ V_(actual) andV_(quad), which are proportional to

$\begin{matrix}{{V_{actual} \propto {\cos \left( \frac{\Phi_{+ 1} + \Phi_{- 1}}{2} \right)}}{{V_{quad} \propto {\sin \left( \frac{\Phi_{+ 1} + \Phi_{- 1}}{2} \right)}},{{so}\mspace{14mu} {that}}}} & (223) \\{\frac{\Phi_{+ 1} + \Phi_{- 1}}{2} = {{{arc}\; {{tg}\left( \frac{V_{quad}}{V_{actual}} \right)}} = {\Phi_{sum}.}}} & (224)\end{matrix}$

The secondary phase shifts are calculated having Φ_(sum) and Φ_(diff) asaccording to

Φ₊₁Φ_(sum)+Φ_(diff)

Φ⁻¹=Φ_(sum)−Φ_(diff)  (225).

With Eq. 16 and Eq.225, the figures of merit M_(+1,−1) ^(D), M_(+1,−1)^(Q) and M_(+1,−1) ^(I) can be calculated.

(c₊₃,c⁻³)-Sampling (PMS)

The two interferograms for (c₊₃,c⁻³)-PMS are given by Eq. 17 and theresulting complex time domain signal S_(+3,−3)(t), according to Eq. 18,is proportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 3},{- 3}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 3},{- 3}}{C_{{+ 3},{- 3}}(t)}}} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 3}(t)} \\{c_{- 3}(t)}\end{bmatrix}}} \\{= {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & \end{bmatrix}}\begin{bmatrix}{- 1} & {- 1} \\{- 1} & 1\end{bmatrix}}} \\{\begin{bmatrix}{\cos \left( {{{+ \alpha}\; t} + \frac{3\pi}{4} + \Phi_{+ 3}} \right)} \\{\cos \left( {{{- \alpha}\; t} + \frac{3\pi}{4} + \Phi_{- 3}} \right)}\end{bmatrix}} \\{= {\sqrt{2}{\cos \left( \frac{\Phi_{+ 3} + \Phi_{- 3}}{2} \right)}}} \\{{^{\lbrack{{{\alpha}\; t} + {\frac{\Phi_{+ 3} - \Phi_{- 3}}{2}}}\rbrack} +}} \\{{\sqrt{2}{\sin \left( \frac{\Phi_{+ 3} + \Phi_{- 3}}{2} \right)}}} \\{{^{\lbrack{{{- {\alpha}}\; t} - {\frac{\Phi_{+ 3} - \Phi_{- 3}}{2}}}\rbrack}.}}\end{matrix} & (226)\end{matrix}$

FT reveals a superposition of absorptive and dispersive components atboth the actual and the quadrature peak positions, with peak intensitiesproportional to

$\cos \left( \frac{\Phi_{+ 3} + \Phi_{- 3}}{2} \right)$ and${\sin \left( \frac{\Phi_{+ 3} + \Phi_{- 3}}{2} \right)},$

respectively. The phase shifts associated with the peaks are given by

$\frac{\Phi_{+ 3} - \Phi_{- 3}}{2}.$

The protocol for obtaining secondary phase shifts from experimentallymeasured peak volumes and the phase shifts of actual and quad peaksinvolves two steps.

Step 1:

Phase correct both the actual and the quad peaks, thereby measuring theassociated phase shifts

$\frac{\Phi_{+ 3} - \Phi_{- 3}}{2}.$

For brevity, we define

$\Phi_{diff} = {\frac{\Phi_{+ 3} - \Phi_{- 3}}{2}.}$

Step 2:

Perform integration on the phase corrected actual and the quad peaks toobtain the peak ‘volumes’ V_(actual) and V_(quad), which areproportional to

$\begin{matrix}{{V_{actual} = {\cos \left( \frac{\Phi_{+ 3} + \Phi_{- 3}}{2} \right)}}{{V_{quad} = {{{\sin \left( \frac{\Phi_{+ 3} + \Phi_{- 3}}{2} \right)}.{so}}\mspace{14mu} {that}}},}} & (227) \\{\frac{\Phi_{+ 3} + \Phi_{- 3}}{2} = {{{arc}\; {{tg}\left( \frac{V_{quad}}{V_{actual}} \right)}} = {\Phi_{sum}.}}} & (228)\end{matrix}$

The secondary phase shifts are calculated using Φ_(sum) and Φ_(diff)according to

Φ₊₃=Φ_(sum)+Φ_(diff)

Φ⁻³=Φ_(sum)−Φ_(diff)  (229).

With Eq.20 and Eq.229, the figures of merit M_(+3,−3) ^(D), M_(+3,−3)^(Q) and M_(+3,−3) ^(I) can be calculated.

0- and π/2-Shifted Combined Forward-Backward Sampling

(c₊₀,c⁻²)-Sampling (PMS)

The two interferograms for (c₊₀,c⁻²)-PMS are given by Eq. 24 and theresulting complex time domain signal S₊₀,c⁻²(t), according to Eq. 25, isproportional to

$\begin{matrix}\begin{matrix}{{{S_{{+ 0},{- 2}}(t)} \propto {\begin{bmatrix}1 & \end{bmatrix}D_{{+ 0},{- 2}}{C_{{+ 0},{- 2}}(t)}}} = {{\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}}\begin{bmatrix}{c_{+ 0}(t)} \\{c_{- 2}(t)}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}c_{+ 0} \\c_{- 2}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & \end{bmatrix}\begin{bmatrix}{\cos \left( {{\alpha \; t} + \Phi_{+ 0}} \right)} \\{\sin \left( {{\alpha \; t} - \Phi_{- 2}} \right)}\end{bmatrix}}} \\{= {\sqrt{2}{\cos \left( \frac{\Phi_{+ 0} + \Phi_{- 2}}{2} \right)}}} \\{{^{\lbrack{{{\alpha}\; t} + {\frac{\Phi_{+ 0} - \Phi_{- 2}}{2}}}\rbrack} -}} \\{{\sqrt{2}{\sin \left( \frac{\Phi_{+ 0} + \Phi_{- 2}}{2} \right)}}} \\{{^{\lbrack{{{- {\alpha}}\; t} - {{({\frac{\pi}{2} - \frac{\Phi_{+ 0} - \Phi_{- 2}}{2}})}}}\rbrack}.}}\end{matrix} & (230)\end{matrix}$

FT reveals a superposition of absorptive and dispersive components atboth the actual and the quadrature peak positions, with peak intensitiesproportional to

${{\cos \left( \frac{\Phi_{+ 0} + \Phi_{- 2}}{2} \right)}\mspace{14mu} {and}\mspace{14mu} {\sin \left( \frac{\Phi_{+ 0} + \Phi_{- 2}}{2} \right)}},$

respectively. The phase shifts associated with the

two peaks differ by π/2, and is

$\frac{\Phi_{+ 0} - \Phi_{- 2}}{2}$

for the actual and by

$\frac{\pi}{2} - \frac{\Phi_{+ 0} - \Phi_{- 2}}{2}$

for the quad peak. The protocol for obtaining secondary phase shiftsfrom experimentally measured peak volumes and phase shifts of the actualand the quad peaks involves two steps.

Step 1:

Phase correct both the actual and the quad peaks and thereby measure theassociated phase shifts

$\frac{\Phi_{+ 0} - \Phi_{- 2}}{2}$

from the required phase correction of actual peak or quad peak aftersubtracting π/2. For brevity, we define

$\frac{\Phi_{+ 0} - \Phi_{- 2}}{2} = {\Phi_{diff}.}$

Step 2:

Perform integration on the phase corrected actual and quad peaks toobtain the peak ‘volumes’ V_(actual) and V_(quad) which are proportionalto

$\begin{matrix}\begin{matrix}{{V_{actual} \propto {\cos \left( \frac{\Phi_{+ 0} + \Phi_{- 2}}{2} \right)}}\mspace{14mu}} \\{{V_{quad} \propto {{\sin \left( \frac{\Phi_{+ 0} + \Phi_{- 2}}{2} \right)}.}}}\end{matrix} & (231) \\{{so}\mspace{14mu} {that}} & \; \\{\frac{\Phi_{+ 0} + \Phi_{- 2}}{2} = {{{arctg}\left( \frac{V_{quad}}{V_{actual}} \right)} = {\Phi_{sum}.}}} & (232)\end{matrix}$

The secondary phase shifts are calculated using Φ_(sum) and Φ_(diff)according to

Φ₊₀=Φ_(sum)+Φ_(diff)

Φ⁻²=Φ_(sum)−Φ_(diff)  (233).

With Eq. 27 and Eq. 233, the figures of merit M_(+0,−2) ^(D), M_(+0,−2)^(Q) and M_(+0,−2) ^(I) can be calculated.

(c⁻⁰,c₊₂)-Sampling (PMS)

The two interferograms for (c⁻⁰,c+2)-PMS are given by Eq.28 and theresulting complex time domain signal S_(−0,+2)(t), according to Eq. 29,is proportional to

$\begin{matrix}{{S_{{- 0},{+ 2}}(t)} \propto \left\lbrack \begin{matrix}1 & \left. i \right\rbrack & {D_{{- 0},{+ 2}}{C_{{- 0},{+ 2}}(t)}}\end{matrix} \right.} \\{= {{\begin{bmatrix}1 & i\end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\begin{bmatrix}{c_{- 0}(t)} \\{c_{+ 2}(t)}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & {- i}\end{bmatrix}\begin{bmatrix}c_{- 0} \\c_{+ 2}\end{bmatrix}}} \\{= {\begin{bmatrix}1 & {- i}\end{bmatrix}\begin{bmatrix}{\cos \left( {{\alpha \; t} - \Phi_{- 0}} \right)} \\{- {\sin \left( {{\alpha \; t} + \Phi_{+ 2}} \right)}}\end{bmatrix}}} \\{{= {{\sqrt{2}{\cos \left( \frac{\Phi_{+ 2} + \Phi_{- 0}}{2} \right)}^{\lbrack{{\; \alpha \; t} + {\frac{\Phi_{+ 2} - \Phi_{- 0}}{2}}}\rbrack}} +}}\mspace{14mu}} \\{{\sqrt{2}{\sin \left( \frac{\Phi_{+ 2} + \Phi_{- 0}}{2} \right)}^{\lbrack{{{- }\; \alpha \; t} + {{({\frac{\pi}{2} - \frac{\Phi_{+ 2} - \Phi_{- 0}}{2}})}}}\rbrack}}}\end{matrix}$

(234).

FT reveals a superposition of absorptive and dispersive components atboth actual and quadrature peak positions, with peak intensitiesproportional to

${{\cos \left( \frac{\Phi_{+ 2} + \Phi_{- 0}}{2} \right)}\mspace{14mu} {and}\mspace{14mu} {\sin \left( \frac{\Phi_{+ 2} + \Phi_{- 0}}{2} \right)}},$

respectively. The phase shifts associated with the peaks differ by π/2and are given by

$\frac{\Phi_{+ 2} - \Phi_{- 0}}{2}$

for the actual and by

$\frac{\pi}{2} - \frac{\Phi_{+ 2} - \Phi_{- 0}}{2}$

for the quad peak. The protocol for obtaining secondary phase shiftsfrom experimentally measured peak volumes and phase shifts of the actualand the quad peaks involves two steps.

Step 1:

Phase correct both the actual and the quad peaks and thereby measure theassociated phase shifts

$\frac{\Phi_{+ 2} - \Phi_{0}}{2}$

from the required phase correction of the actual peak or from the phasecorrection value of the quad peak after subtracting π/2. For brevity, wedefine

$\Phi_{diff} = {\frac{\Phi_{+ 2} - \Phi_{0}}{2}.}$

Step 2:

Perform integration on the phase corrected actual and quad peaks toobtain the volumes V_(actual) and V_(quad), which are proportional to

$\begin{matrix}\begin{matrix}{{V_{actual} \propto {\cos \left( \frac{\Phi_{+ 2} - \Phi_{0}}{2} \right)}}\mspace{14mu}} \\{{V_{quad} \propto {{\sin \left( \frac{\Phi_{+ 2} + \Phi_{0}}{2} \right)}.}}}\end{matrix} & (235) \\{{so}\mspace{14mu} {that}} & \; \\{\frac{\Phi_{+ 2} + \Phi_{0}}{2} = {{{arctg}\left( \frac{V_{quad}}{V_{actual}} \right)} = {\Phi_{sum}.}}} & (236)\end{matrix}$

The secondary phase shifts are calculated using Φ_(sum) and Φ_(diff)according to

Φ⁻⁰=Φ_(sum)−Φ_(diff)

Φ₊₂=Φ_(sum)+Φ_(diff)  (237).

With Eq. 27 and Eq. 237, the figures of merit M_(+0,−2) ^(D), M_(+0,−2)^(Q) and M_(+0,−2) ^(I) can be calculated.

Results

(c₊₁,c⁻¹)-, (c₊₃,c⁻³)-, (c₊₀,c⁻²)- and (c⁻⁰,c₊₂)-PMS was employed toacquire aliphatic constant-time 2D [¹³C,¹H]-HSQC spectra recorded for a2 mM solution of ¹⁵N/¹³C-labeled phenylalanine (FIGS. 8A-D). Therequired interferograms were acquired with different delays for delayedacquisition, so that different secondary phase shifts are obtained forthe time domain signals in the interferograms. FIGS. 8A-D show crosssections taken along ω₁(¹³C) at the ¹H-¹³C^(α) signal from spectrarecorded with (c₊₁,c⁻¹)-sampling (FIG. 8A), (c₊₃,c⁻³)-sampling (FIG.8B), (c₊₀,c⁻²)-sampling (FIG. 8C) and (c⁻⁰,c₊₂)-sampling (FIG. 8D). Thepeaks exhibit mixed phase due to differences of secondary phase shifts(Table 10). The quad peaks for the (c₊₀,c⁻²)- and (c⁻⁰,c₊₂)-sampledspectra are shown after a π/2 zero-order phase correction was applied.The secondary phase shifts were measured using the protocols providedabove. Table 10 provides a comparison of predicted and measuredsecondary phase shifts, revealing the expected agreement between theoryand experiment.

TABLE 10 Comparison of predicted and measured secondary phase shifts^(a)Secondary Secondary phase Measured phase shifts due to delayedsecondary, phase Sampling scheme shifts acquisition^(b) shifts^(c) (c₊₁,c⁻¹) − (PMS) Φ₊₁ 29.0° 29.9° Φ⁻¹ 20.9° 20.7° (c₊₃, c⁻³) − (PMS) Φ₊₃24.9° 26.5° Φ⁻³ 16.9° 16.7° (c₊₀, c⁻²) − (PMS) Φ₊₀ 29.0° 28.2° Φ⁻² 16.9°18.2° (c⁻⁰, c₊₂) − (PMS) Φ⁻⁰ 24.9° 24.3° Φ₊₂ 20.9° 20.8° ^(a)Sample:¹⁵N, ¹³C-labeled phenylalanine (2 mM concentration at pH 6.5). Allspectra were recorded at 25° C. on a Varian 500 MHz NMR spectrometer.^(b)Delays Δt for delayed acquisition were set for generating secondaryphase shifts according to Φ = 2π*(Ω − Ω_(carrier))*Δt, where (Ω −Ω_(carrier)) represents the offset relative to the carrier frequency inHz. ^(c)Eqs. 225, 229, 233, 237 were used to obtain the secondary phaseshifts.

The measurement of the secondary phase shifts enables one to calculatethe figures of merit for the different sampling schemes (Table 11) toquantitatively assess the different sampling schemes.

TABLE 11 Figures of merit^(a) M^(D), M^(Q) and M^(I) for the samplingschemes of Table 10 corresponding DMPS sampling schemes M^(D) M^(Q)M^(I) (c₊₁, c⁻¹) − (PMS) 0.92 0.68 0.90 (c₊₃, c⁻³) − (PMS) 0.92 0.720.93 (c₊₁, c⁻¹, c₊₃, c⁻³) − (DPMS) 0.92 0.94 0.91 (c₊₀, c−₂) − (PMS)0.92 0.70 0.92 (c⁻⁰, c₊₂) − (PMS) 0.97 0.70 0.92 (c₊₀, c⁻², c⁻⁰, c₊₂) −(PMS) 0.94 0.97 0.92 ^(a)Obtained with Eqs. 16, 20, 27, 31.

Although preferred embodiments have been depicted and described indetail herein, it will be apparent to those skilled in the relevant artthat various modifications, additions, substitutions, and the like canbe made without departing from the spirit of the invention and these aretherefore considered to be within the scope of the invention as definedin the claims which follow.

1. A method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by use of forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts, said method comprising: providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N; sampling a time domain modulation in a phase-sensitive manner in each selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift α_(j) in both a forward and backward manner to obtain two interferograms for each time domain dimension t_(j) defining the vector $\begin{matrix} {{C_{j,\psi_{j}}\left( t_{j} \right)}:=\begin{bmatrix} {I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}} \end{bmatrix}} \\ {{= \begin{bmatrix} {I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}} \end{bmatrix}},{wherein}} \end{matrix}$ I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j,δ) _(j) ⁻ are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j) are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[and the cases {ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ are secondary phase shifts; multiplying each said vectors C_(j,ψ) _(j) (t_(j)) with a D-matrix defined as $D_{j} = \begin{bmatrix} {\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\ {- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)} \end{bmatrix}$ and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·D_(j)·C_(j,ψ) _(j) (t_(j)) under conditions effective to create a complex time domain of said selected m dimensions according to ${\underset{j}{\otimes}{Q \cdot D_{j} \cdot {C_{j,\psi_{j}}\left( t_{j} \right)}}};{and}$ transforming said complex time domain into frequency domain by use of an operator O under conditions effective to measure the values of I_(j,ψ) _(j) ⁺, I_(j,ψ) _(j,δ) _(j) ⁻, Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ said frequency domain.
 2. The method according to claim 1, wherein said operator is a linear operator L.
 3. The method according to claim 2, wherein said linear operator is the Fourier transformation operator F.
 4. The method according to claim 2, wherein said vectors C_(j,ψ) _(j) (t_(j)) are transformed into frequency domain using said linear operator L under conditions effective to yield frequency domain vector L[C_(j,ψ) _(j) (t_(j))], and said multiplying comprises: multiplying L[C_(j,ψ) _(j) (t_(j))] with said matrix D_(j) and said vector Q according to Q·D_(j)·L[C_(j,ψ) _(j) (t_(j))] under conditions effective to generate said frequency domain according to $\underset{j}{\otimes}{Q \cdot D_{j} \cdot {{L\left\lbrack {C_{j,\psi_{j}}\left( t_{j} \right)} \right\rbrack}.}}$
 5. The method according to claim 1, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=π/4 and δ_(j)=0} or {Ψ_(j)=3π/4 and δ_(j)=0}.
 6. The method according to claim 5, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain are purely absorptive and devoid of dispersive components.
 7. The method according to claim 1, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=0 and δ_(j)=π/2}, or {Ψ_(j)=π/2 and δ_(j)=3π/2}.
 8. The method according to claim 7, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain located at (α₁, α₂, . . . α_(m)) are purely absorptive and devoid of dispersive components.
 9. The method of claim 1, wherein m′ dimensions of said selected m dimensions with m′≦m are jointly sampled according to t=t₁/κ₁=t₂/κ₂= . . . =t_(m)′/κ_(m)′, wherein κ_(m)′ are scaling factors for time evolution in the jointly sampled dimensions, under conditions effective to conduct a G-matrix Fourier Transformation NMR experiment, wherein said multiplication with said vector Q is replaced for K=m′−1 of the m′ jointly sampled dimensions by multiplication with matrix G defined as $G = {\begin{bmatrix} 1 &  \\ 1 & {- } \end{bmatrix}.}$
 10. The method according to claim 9, wherein for all j: I_(j,ψ) _(j+) =I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=π/4 and δ_(j)=0} or {Ψ_(j)=3π/4 and δ_(j)=0}.
 11. The method according to claim 10, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) =, so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components.
 12. The method according to claim 9, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=0 and δ_(j)=π/2}, or {Ψ_(j)=π/2 and δ_(j)=3π/2}.
 13. The method according to claim 12, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain located at linear combinations of (α₁, α₂, . . . α_(m)) for a given sub-spectrum of the G-matrix Fourier Transformation NMR experiment are purely absorptive and devoid of dispersive signal components.
 14. The method according to claim 1, wherein said secondary phase shifts Φ_(j,ψ) _(j) ⁺, and Φ_(j,ψ) _(j,δ) _(j) ⁻ encode NMR parameters other than said chemical shifts α_(j).
 15. The method according to claim 1, wherein said sampling a time domain modulation is combined with preservation of equivalent pathways for sensitivity enhancement.
 16. The method according to claim 1, wherein said NMR experiment is a TROSY NMR experiment.
 17. The method according to claim 1, wherein said sampling a time domain modulation is accomplished by use of simultaneous phase cycled NMR.
 18. The method according to claim 1, wherein said interferograms are obtained by recording P- and N-type time domain by use of pulsed magnetic field gradients followed by linear combination effective to generate said interferograms.
 19. A method of optimizing an N-dimensional nuclear magnetic resonance (NMR) experiment comprising: measuring values of secondary phase shifts Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ in a first N-dimensional NMR experiment according to the method of claim 1; identifying an origin of the secondary phase shifts Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ in the first N-dimensional NMR experiment; and modifying a radio frequency pulse scheme of a second N-dimensional NMR experiment under conditions effective to at least partially eliminate the origin of the secondary phase shifts Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ and at least partially eliminate said secondary phase shifts Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻.
 20. A method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts, said method comprising: providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N; sampling a time domain modulation in a phase-sensitive manner in each said selected dimension jε=[1,2, . . . , m] arising from time evolution of chemical shift α_(j) in both a forward and backward manner to obtain two interferograms for each time domain dimension t_(j) defining the vector $\begin{matrix} \begin{matrix} {{C_{j,\psi_{j}}\left( t_{j} \right)}:=\begin{bmatrix} {I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j}\;,\delta_{j}}^{-}\left( t_{j} \right)}} \end{bmatrix}} \\ {{= \begin{bmatrix} {I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\ {I_{j,\psi^{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}} \end{bmatrix}},} \end{matrix} & \; \\ {wherein} & \; \end{matrix}$ I_(j,ψ) _(j) ⁺ and I_(j,ψ) _(j,δ) _(j) ⁻ are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j) are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[and the cases {ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ are secondary phase shifts; multiplying each said vectors C_(j,ψ) _(j) (t_(j)) with a D-matrix defined as $D_{j} = \begin{bmatrix} {\sin \left( {\psi_{j} + \delta_{j}} \right)} & {\sin \left( \psi_{j} \right)} \\ {- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)} \end{bmatrix}$ and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·D_(j)·C_(j,ψ) _(j) (t_(j)) under conditions effective to create a complex time domain of said selected m dimensions according to ${\underset{j}{\otimes}{Q \cdot D_{j} \cdot {C_{j,\psi_{j}}\left( t_{j} \right)}}};$ repeating said selecting, said sampling and said multiplying (2^(m)−1)-times, thereby sampling the m dimensions with all 2^(m) possible permutations resulting from selecting for each dimension j either Ψ_(j) or Ψ_(j)+π/2, with δ_(j) being incremented by either 0 or π, thereby yielding 2^(m) complex time domains; linearly combining said 2^(m) complex time domains; and transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to at least partially cancel dispersive and quadrature image peak components arising from Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ in said frequency domain.
 21. The method according to claim 20, wherein said transforming comprises measuring the values of I_(j,ψ) _(j) ⁺, I_(j,ψ) _(j,δ) _(j) ⁻ Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ in said frequency domain.
 22. The method according to claim 20, wherein said operator is a linear operator L.
 23. The method according to claim 22, wherein said linear operator is the Fourier transformation operator F.
 24. The method according to claim 22, wherein said vectors C_(j,ψ) _(j) (t_(j)) are transformed into frequency domain using said linear operator L under conditions effective to yield frequency domain vector L[C_(j,ψ) _(j) (t_(j))], and said multiplying comprises: multiplying L[C_(j,ψ) _(j) (t_(j))] with said matrix D_(j) and said vector Q according to Q·D_(j)·L[C_(j,ψ) _(j) (t_(j))] under conditions effective to generate said frequency domain according to $\underset{j}{\otimes}{Q \cdot D_{j} \cdot {{L\left\lbrack {C_{j,\psi_{j}}\left( t_{j} \right)} \right\rbrack}.}}$
 25. The method according to claim 20, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=π/4 and δ_(j)=0} or {Ψ_(j)=3π/4 and δ_(j)=0}.
 26. The method according to claim 25, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain are purely absorptive and devoid of dispersive components.
 27. The method according to claim 25, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j), so that signals in said frequency domain are purely absorptive and devoid of dispersive components, and quadrature image peaks in said frequency domain are cancelled.
 28. The method according to claim 20, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=0 and δ_(j)=π/2}, or {Ψ_(j)=π/2 and δ_(j)=3π/2}.
 29. The method according to claim 28, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain located at (α₁, α₂, . . . α_(m)) are purely absorptive and devoid of dispersive components.
 30. The method according to claim 28, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j), so that signals in said frequency domain located at (α₁, α₂, . . . , α_(m)) are purely absorptive and devoid of dispersive components, and quadrature image peaks in said frequency domain are cancelled.
 31. The method according to claim 20, wherein m′ dimensions of said selected m dimensions with m′≦m are jointly sampled according to t=t₁/κ₁=t₂/κ₂= . . . =t_(m)′/κ_(m)′, wherein κ_(m)′ are scaling factors for the time evolution in the jointly sampled dimensions under conditions effective to conduct a G-matrix Fourier Transformation NMR experiment, wherein said multiplication with said vector Q is replaced for K=m′−1 of the m′ jointly sampled dimensions by multiplication with matrix G defined as $G = {\begin{bmatrix} 1 &  \\ 1 & {- } \end{bmatrix}.}$
 32. The method according to claim 31, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=π/4 and δ_(j)=0} or {Ψ_(j)=3π/4 and δ_(j)=0}.
 33. The method according to claim 32, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components.
 34. The method according to claim 32, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j), so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components, and quadrature image and cross talk peaks in said frequency domain are cancelled.
 35. The method according to claim 31, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁻, and {Ψ_(j)=0 and δ_(j)=π/2}, or {Ψ_(j)=π/2 and δ_(j)=3π/2}.
 36. The method according to claim 35, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain located at linear combinations of (α₁, α₂, . . . α_(m)) for a given sub-spectrum of the GFT NMR experiment are purely absorptive and devoid of dispersive signal components.
 37. The method according to claim 35, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j), so that signals in said frequency domain located at linear combinations of (α₁, α₂, . . . α_(m)) for a given sub-spectrum of the GFT NMR experiment are purely absorptive and devoid of dispersive signal components, and quadrature image and cross talk peaks in said frequency domain are cancelled.
 38. The method according to claim 20, wherein said permutation of said secondary phase shifts is concatenated with execution of a radio-frequency phase cycle.
 39. The method according to claim 38, wherein said NMR experiment is conducted under conditions of magic angle spinning of said sample.
 40. The method according to claim 38, wherein said phase cycle is executed to suppress signals arising from axial magnetization.
 41. The method according to claim 38, wherein said phase cycle is executed to reduce signal arising from solvent.
 42. The method according to claim 20, wherein said sampling a time domain modulation is combined with preservation of equivalent pathways for sensitivity enhancement.
 43. The method according to claim 20, wherein said NMR experiment is a TROSY NMR experiment.
 44. The method according to claim 20, wherein said sampling a time domain modulation is accomplished by use of simultaneous phase cycled NMR.
 45. The method according to claim 20, wherein said interferograms are obtained by recording P- and N-type time domain by use of pulsed magnetic field gradients followed by linear combination effective to generate said interferograms.
 46. A method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts, said method comprising: providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N; sampling twice a time domain modulation in a phase-sensitive manner in each said selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift α_(j), once in a forward manner to obtain two interferograms for each time domain dimension t_(j) defining the vector $\begin{matrix} {{C_{j,\psi_{j}}^{+}\left( t_{j} \right)} = \begin{bmatrix} {I_{j,\psi_{j}}^{+}{c_{\psi_{j}}^{+}\left( t_{j} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{+}{c_{\psi_{j},\delta_{j}}^{+}\left( t_{j} \right)}} \end{bmatrix}} \\ {{= \begin{bmatrix} {I_{j,\psi_{j}}^{+}{\cos \left( {\psi_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{+}} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{+}{\cos \left( {\psi_{j} + \delta_{j} + {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{+}} \right)}} \end{bmatrix}},} \end{matrix}$ and once in a backward manner to obtain two interferograms for each time domain dimension t_(j) defining the vector $\begin{matrix} {{C_{j,\psi_{j}}^{-}\left( t_{j} \right)}:=\begin{bmatrix} {I_{j,\psi_{j}}^{-}{c_{\psi_{j}}^{-}\left( t_{j} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{-}{c_{\psi_{j},\delta_{j}}^{-}\left( t_{j} \right)}} \end{bmatrix}} \\ {{= \begin{bmatrix} {I_{j,\psi_{j}}^{-}{\cos \left( {\psi_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j}}^{-}} \right)}} \\ {I_{j,\psi_{j},\delta_{j}}^{-}{\cos \left( {\psi_{j} + \delta_{j} - {\alpha_{j}t_{j}} + \Phi_{j,\psi_{j},\delta_{j}}^{-}} \right)}} \end{bmatrix}},} \end{matrix}$ wherein I_(j,ψ) _(j) ⁺, I_(j,ψ) _(j,δ) _(j) ⁺, I_(j,ψ) _(j) ⁻ and I_(j,ψ) _(j,δ) _(j) ⁻ are amplitudes, Ψ_(j) and Ψ_(j)+δ_(j) are primary phase shifts with Ψ_(j), δ_(j)ε[0,2π[, and Φ_(j,ψ) _(j) ⁺, Φ_(j,ψ) _(j,δ) _(j) ⁺, Φ_(j,ψ) _(j) ⁻ and Φ_(j,ψ) _(j,δ) _(j) ⁻ are secondary phase shifts; multiplying each said vector C_(j,ψ) _(j) ⁺(t_(j)) with a D-matrix defined as $D_{j}^{+} = \begin{bmatrix} {\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\ {\cos \left( {\psi_{j} + \delta_{j}} \right)} & {- {\cos \left( \psi_{j} \right)}} \end{bmatrix}$ and each said vector C_(j,ψ) _(j) ⁻(t_(j)) with a D-matrix defined as ${D_{j}^{-} = \begin{bmatrix} {\sin \left( {\psi_{j} + \delta_{j}} \right)} & {- {\sin \left( \psi_{j} \right)}} \\ {- {\cos \left( {\psi_{j} + \delta_{j}} \right)}} & {\cos \left( \psi_{j} \right)} \end{bmatrix}};$ multiplying the said products D_(j) ⁺·C_(j,ψ) _(j) ⁺(t_(j)) and D_(j) ⁻C_(j,ψ) _(j) ⁻(t_(j)) with a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·D_(j) ⁺·C_(j,ψ) _(j) ⁺(t_(j)) and Q·D_(j) ⁻·C_(j,ψ) _(j) ⁻(t_(j)) under conditions effective to create a complex time domain of said selected m dimensions according to $\underset{j}{\otimes}{Q \cdot D_{j}^{+} \cdot {C_{j,\psi_{j}}^{+}\left( t_{j} \right)}}$ ${{{and}\underset{j}{\otimes}Q} \cdot D_{j}^{-} \cdot {C_{j,\psi_{j}}^{-}\left( t_{j} \right)}};$ repeating said selecting, said phase-sensitive sampling twice and said multiplying (2^(m)−2)-times, thereby sampling said m dimensions with all 2^(m) possible permutations resulting from selecting for each dimension j either phase-sensitive forward or backward sampling according to C_(j,ψ) _(j) ⁺(t_(j)) or C_(j,ψ) _(j) ⁻(t_(j)); linearly combining said 2^(m) complex time domains; and transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to at least partially cancel dispersive and quadrature image peak components arising from Φ_(j,ψ) _(j) ⁺, Φ_(j,ψ) _(j,δ) _(j) ⁺, Φ_(j,ψ) _(j) ⁻ and Φ_(j,ψ) _(j,δ) _(j) ⁻ in said frequency domain.
 47. The method according to claim 46, wherein said sampling in a phase sensitive manner comprises excluding {ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0,
 1. 48. The method according to claim 46, wherein said sampling in a phase sensitive manner for {ψ_(j)=nπ/2 and δ_(j)=mπ} with n=0, 1, 2, 3 and m=0, 1 comprises applying time-proportional phase incrementation of radio-frequency pulse or receiver phases.
 49. The method according to claim 46, wherein said transforming comprises measuring the values of I_(j,ψ) _(j) ⁺, I_(j,ψ) _(j,δ) _(j) ⁻, Φ_(j,ψ) _(j) ⁺ and Φ_(j,ψ) _(j,δ) _(j) ⁻ in said frequency domain.
 50. The method according to claim 46, wherein said operator is a linear operator L.
 51. The method according to claim 50, wherein said linear operator is the Fourier transformation operator F.
 52. The method according to claim 50, wherein said vectors C_(j,ψ) _(j) ⁺(t_(j)) and C_(j,ψ) _(j) ⁻(t_(j)) are transformed into frequency domain using said linear operator L under conditions effective to yield frequency domain vectors L[C_(j,ψ) _(j) ⁺(t_(j))] and L[C_(j,ψ) _(j) ⁻(t_(j))], and said multiplying comprises: multiplying said vector L[C_(j,ψ) _(j) ⁺(t_(j))] with said matrix D_(j) ⁺ and said vector L [C_(j,ψ) _(j) ⁻(t_(j))] with said matrix D_(j) ⁻; and multiplying said products D_(j) ⁺·L [C_(j,ψ) _(j) ⁺(t_(j))] and D_(j) ⁻·L [C_(j,ψ) _(j) ⁻(t_(j))] with said vector Q according to Q·D_(j) ⁺·L[C_(j,ψ) _(j) ⁺(t_(j))] and Q·D_(j) ⁻·L[C_(j,ψ) _(j) ⁻(t_(j))] under conditions effective to generate said frequency domain according to $\underset{j}{\otimes}{Q \cdot D_{j}^{+} \cdot {L\left\lbrack {C_{j,\psi_{j}}^{+}\left( t_{j} \right)} \right\rbrack}}$ ${{and}\underset{j}{\otimes}Q} \cdot D_{j}^{-} \cdot {{L\left\lbrack {C_{j,\psi_{j}}^{-}\left( t_{j} \right)} \right\rbrack}.}$
 53. The method according to claim 46, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁺=I_(j,ψ) _(j) ⁻=I_(j,ψ) _(j,δ) _(j) ⁻, and {ψ_(j)=0 and δ_(j)=π/2}.
 54. The method according to claim 53, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j) ⁻=Φ_(j,ψ) _(j) and Φ_(j,ψ) _(j,δ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain are purely absorptive and devoid of dispersive components.
 55. The method according to claim 53, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j), so that signals in said frequency domain are purely absorptive and devoid of dispersive components, and quadrature image peaks in said frequency domain are entirely cancelled.
 56. The method according to claim 46, wherein m′ dimensions of said selected m dimensions with m′≦m are jointly sampled according to t=t₁/κ₁=t₂/κ₂= . . . =t_(m)′/κ_(m)′, wherein κ_(m)′ are the scaling factors for the time evolution in the jointly sampled dimensions, effective to conduct a G-matrix Fourier Transformation NMR experiment, wherein said multiplication with said vector Q is replaced for K=m′−1 of the m′jointly sampled dimensions by multiplication with matrix G defined as $G = {\begin{bmatrix} 1 &  \\ 1 & {- } \end{bmatrix}.}$
 57. The method according to claim 56, wherein for all j: I_(j,ψ) _(j) ⁺=I_(j,ψ) _(j,δ) _(j) ⁺=I_(j,ψ) _(j) ⁻=I_(j,ψ) _(j,δ) _(j) ⁻, and {ψ_(j)=0 and δ_(j)=π/2}.
 58. The method according to claim 57, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j) ⁻=Φ_(j,ψ) _(j) and Φ_(j,ψ) _(j,δ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=I_(j,ψ) _(j,δ) _(j) , so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components.
 59. The method according to claim 57, wherein Φ_(j,ψ) _(j) ⁺=Φ_(j,ψ) _(j) ⁻=Φ_(j,ψ) _(j,δ) _(j) ⁺=Φ_(j,ψ) _(j,δ) _(j) ⁻=Φ_(j), so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components, and quadrature image and cross talk peaks in said frequency domain are entirely cancelled.
 60. The method according to claim 46, wherein said permutation of said secondary phase shifts is concatenated with execution of a radio-frequency phase cycle.
 61. The method according to claim 60, wherein said NMR experiment is conducted under conditions of magic angle spinning of said sample.
 62. The method according to claim 60, wherein said phase cycle is executed to suppress signals arising from axial magnetization.
 63. The method according to claim 60, wherein said phase cycle is executed to reduce signal arising from solvent.
 64. The method according to claim 46, wherein said sampling a time domain modulation is combined with preservation of equivalent pathways for sensitivity enhancement.
 65. The method according to claim 46, wherein said NMR experiment is a TROSY NMR experiment.
 66. The method according to claim 46, wherein said sampling a time domain modulation is accomplished by use of simultaneous phase cycled NMR.
 67. The method according to claim 46, wherein said interferograms are obtained by recording P- and N-type time domain by use of pulsed magnetic field gradients followed by linear combination effective to generate said interferograms. 